The kid sitting across from me looked like everyone else in my pre-calculus class. He had a mop of brown hair on his head and was wearing an AC-DC shirt. He'd been assigned as my partner for a class project. I hadn't been looking forward to this. Historically speaking, I always end up doing most of the work on those. And then this kid, before he's even set a word to me, hands me a business card. This was a novelty unto itself, but then I took the thing and it said Nolan Beckett, Paranormal Detective. The project can wait. So I said, Paranormal Detective, does that require any qualifications? Only that you're willing to look for stuff in places that other people aren't. The kid said, he extended his hand. I'm Nolan, by the way. Yeah, I saw the card. I said, I took his hand and shook it. The name's Joseph.  Nolan said, please don't.  Let's go with that. And then Nolan just reached into his backpack and pulled out his pre-calculus notebook. And he starts to do research for our project like we didn't just have that conversation. Wait, we can't just leave it at that. I said, you don't just tell someone you're a Paranormal Detective and then just end the conversation there. I do, Nolan said. What I've learned is that no one believes me except for the people who hire me. Hit me. I said, Nolan sighed and closed his notebook. Okay, let me give you the short list of what I've done. I've shot a troupe of cabra with a shotgun at point blank range. I have done pest control for a plane infested with gremlins and I burned down a possessed barn. I don't allow my face to betray any emotion. It was the greatest test of will I'd faced in my life. You don't believe me? Nolan said, I didn't say that. I replied, I can tell.  If I showed photos, you would say they can be faked. There's just no convincing regular people about what I do. Okay, so why hand me the business card? I like to see how people react. With most people it's entertaining, but you were pretty boring. Nolan looked back down at his notebook, continuing to work. I didn't expect a kid who carries around business cards that say he's a Paranormal Detective to be so studious. At this point, I faced a choice. I could leave this alone, complete the project, never talk to this guy again, and then maybe laugh about this episode years later over some beers with some friends. Or, what if I were to tag along on one of your cases? I said, Nolan looked up from his notebook. You're serious? I nodded. No one has ever offered to do that before. Nolan grinned. It just so happens, I am working a case right now. Nolan closed his notebook. I must warn you though, if this case turns out like I think it will, things could get dangerous. Dangerous is my middle name. I replied. I hurried out that, Nolan said. Later that day, I told my parents I was going to hang out with some friends, which wasn't far from the truth. I don't get out much, so they were happy that I was being social at all and didn't ask many questions. I think they would have stopped me if they knew who I was hanging out with. Nolan had given me an address and time to meet him at. As I pulled up to the address, I saw that it was a seemingly normal two-story suburban house. In the driveway, there was a blue Chevy Tahoe. And then there was what I can only presume to be Nolan's car. It was a dirt brown Lincoln Continental that looked like it was being held together with prayer. Ducktape was covering one of the windows and was holding the back bumper up. It was scratched in what must have been a dozen different places and the windshield had two chips in it. Now the reason I assumed it was Nolan's was because he was leaning up against it looking at his phone.  I parked in the street and walked up to him. Well, I'm here. Now what? Okay, now first take these because we will need them if what I think is happening here happens. Nolan reached into the back seat of his car through the open car window and pulled out some road flares. He handed me two and kept the other two for himself. And what is happening here? I asked, come on, I'll brief you inside. Me and Nolan sat down at the kitchen table inside the house. Nolan had apparently been given a spare key by the owner. Okay, so the owner of this place was here noises in his basement. This was way louder than any rat or anything. And when he went down to investigate, he found this. Nolan pulled two Polaroid photos out of his pocket and put them on the table.  The other picture was of some kind of deep freezer. It was open and the meat that had been stored inside was strewn across the concrete floor. Now, he didn't think the cops would do anything with the case this weird show he called me. So you got this thing solved? I asked. I think I do. If I'm right, then we just have to wait for the sun to set and then we can kill whatever this thing is. Wait. Kill? Yes. But trust me, if we don't get rid of this thing, then no one else is going to be able to. So now we just wait? Yes. We waited for about an hour for the sun to set. Nolan raided the owner's fridge, but I just sat on the couch playing games on my phone.  Nolan turned to look at me. You sure you want that proof? I thought about it for a moment. And then I said, yeah, I've come too far now to not get a little proof. Nolan grinned. Dashed a spirit. Then he opened the basement door. The wooden steps descended into the darkness of the basement. There was a light switch to the left of the door. I tried flipping it, but it had no effect that that thing must have broken the lights. Nolan said he reached into his pocket and pulled out a flashlight, shining it into the darkness. He began to descend the steps and I followed him. We reached the basement. There were cardboard boxes and crates piled high all around us, creating a maze. There, look, Nolan said. Nolan swung the flashlight to his left. I followed the beam. And there it was, the hole in the concrete from the photo Nolan showed me. I walked over to it. Its edges and sides were rough like the hole had been hurriedly made. What could make this? I said, that's what we're here to find out. Suddenly there was a skittering sound coming from behind all the boxes and crates. Nolan and I stopped and began looking all around us. Joel Nolan said, get your flair ready.  I took one of my flares and struggled to do the same. Suddenly something dropped from the ceiling onto Nolan. He dropped his flair and flashlight. I looked up from my flair and I saw something hunched over him. The thing picked him up and slammed him against the concrete wall. I grabbed the flashlight Nolan had dropped and pointed its beam at whatever had attacked him. It was bipedal, about six feet tall. It had see-through skin. I could see its organs and veins and I saw them throbbing as they did their job of keeping this thing alive. Its arms were longer than they should have been and its hands had four claws that were a foot long each. The claws almost looked like they were made out of bone or something like it. The thing turned to face me. It had huge eyes that were completely black and its teeth were long and sharp. As I pointed the flashlight at its face, the thing roared at me, revealing that it actually had row upon row of sharp teeth. This thing had Nolan pinned against the wall and I ran. I booked it for the stairs and was almost at the top when I heard, Joe. I stopped. I couldn't leave Nolan down there, but what could I do? I turned back. The thing wasn't following me. Why? When I remembered its eyes, all black. I remembered learning about that kind of thing in science class. Some species of animals have developed eyes that are essentially all pupil to absorb as much light as possible. That could be why it had roared at me when I shone the flashlight at it and why it had taken out Nolan who had a flare first. Suddenly I had a plan. I started descending the stairs again. As I did, I finally got my flare working. When I reached the bottom, I saw that the thing still had Nolan pinned against the wall. I looked at the floor. Nolan's flare was still burning. I grabbed it. Hey ugly, I yelled out. The thing turned to face me again and I saw it flinch from the light of the flares. Then I ran straight at it. I sprinted right towards it and before it could react, I jabbed the flares into its eyes. The thing let out the most blood curdling screech I'd ever heard and then it ran towards the hole in the wall. It frantically crawled inside of it and then it was gone before I could do so much as get another decent look at it. I looked at Nolan. He was slumped against the wall, taking deep breaths, but he was grinning like nothing was wrong. Where? Nolan said. D-do you believe me now? Yeah. Yeah. I believed him. Only question was, what do I do now? Nolan sat on a chair in the kitchen. He held a ziplock filled with ice to the back of his head, where a lump was forming from when that, that monster, slammed him into the concrete wall of the basement. I found it a miracle that the wound had only bled a little bit. At this rate, he'd be only left with a nasty bruise. I sat in a chair across the table from him. We were looking at each other, but not saying anything. An awkward silence hung in the air between us.  Apparently Nolan had the answer to that question. So, he said, was that enough proof for you, Mr. Joseph? Yeah. Yeah, I'd say that was definitely enough proof for something, all right? What as great? Nolan clapped the hand that wasn't holding the ziplock on his knee and stood up. So I assumed that means you're done with me and this whole supernatural business, yeah? I opened my mouth to respond, but then I paused. I'd already experienced something that would make most people question everything. I stabbed the two road flares into the eyes of a monster that had come straight out of the imagination of dudes like Lovecraft and Stephen King. Wouldn't the smart thing to do beat a dip out now? And yet, nah. I said as I grinned, either because of the insanity of the situation or my own cockiness. That seemed like light work. Nolan's face betrayed his confusion. He stood silent for a moment, seemingly processing what I just said.  So who am I to judge? So what's next? I asked, well, if you're up to it, there's another client I was hoping to help today. It'd be a shame if I had to cancel. I called my parents and let them know that the hangout had turned into me crashing for the night at my friend's place. They seemed nervous at first, but they softened towards the idea and eventually agreed on me staying the night. It was a Friday after all, so I wouldn't have to wake up in the morning for anything.  When I'd pointed out that this would look terrible, Nolan reached into his jacket pocket and pulled out a beanie. Do you just carry that around? Yes, this kind of thing happened quite a lot. After we'd done our best to conceal the Ziploc full of ice, we walked out of the house and to our cars. Wait, wait, Nolan said as we walked out, can can you help me get the car started? I paused for a moment and then decided asking as to why starting Nolan's car was a two man job wouldn't hold a candle to the myriad of questions I could be asking him right now. So I just said, sure, Nolan instructed me to take a seat on the passenger side while he got into the driver's seat. Okay, so when the engine turns, I need you to stop on a red X, Nolan said. I looked down and sure enough, an X made with red tape was on the floor of the car. I raised my foot in preparation. Actually, it worked better if you use both feet, Nolan said. I raised both my feet in preparation, Nolan turned the keys and as the engine turned I stomped down as hard as I could on the red X. The old Lincoln Continental roared to life, its engine making the whole car rumble slightly. Thanks, Nolan said, getting that started is way easier with a second person. What do you do if you don't have a second person? I asked, Nolan pointed his thumb over his shoulder at the back seat. I turned my head and laying across the back seat was a wooden baseball bat. I got out of the Continental and got into my own car. Nolan pulled his land boat out of the driveway and went down the street and I followed him to the next case. We ended up driving out to a house that was a little ways into the country. It was a 30 minute drive from the Ring Road that went around our city so there was nothing but fields all around us. Me and Nolan parked our cars in the dirt driveway. We hopped out and walked up to the door. The house looked fairly normal. It was a one story that could have been plucked out of the suburbs. We were just in Nolan walked right up to the front door and knocked. The door was opened by an old man. I mean he was old, but not old old. I would have put his age at somewhere around the early 60s.  He was smiling when he opened the door. Well hello there. How can I help you two boys? Hello sir. My name is Nolan Beckett and this is my associate Joe. We understand you are facing a certain situation. The smile quickly faded from the man's face. Oh, so you're here about that. The man looked around as though someone would have seen us out in the middle of the country. Well then, come in. The man stepped aside and me and Nolan walked into the house. We sat on a couch in the man's living room. The man walked in carrying drinks he'd gotten from the kitchen. I hope you two like soda because it was either that or beer. The man said he carried two Dr. Peppers and a Heineken in his hands. He handed me and Nolan the Dr. Peppers and he kept the Heineken to himself. He sat in a recliner that sat beside the couch. Thank you, sir. Nolan said he cracked the tab on the Dr. Pepper and took a sip. I just held mine in my hands. All right, so let me just start off with this. The man said I was a doctor for about 40 years, I believe in science. In fact, a voice on the back of my head is telling me the two of you might be frauds looking to con some money out of an old man. No offense, doc, but you should probably see a doc if you're hearing voices. Nolan said he chuckled at his own joke. The man and me didn't laugh though. So Nolan just took another sip from his Dr. Pepper. Like I was saying, the doctor continued, I believe in things like the scientific process, but there's no way to use science to explain what happened a couple nights ago. The doctor stood up and motioned for us to stand up as well. We stood up and followed him to the door to his backyard. He put his hand on the door knob and then looked to us. Just a fair warning, this is going to be a lot. The doctor swung open the door to a lot of dead chickens, and I mean a lot of dead chickens like an inordinate amount. The backyard was fairly large, but I could still see the barbed wire fence off in the distance. It was a chicken coop a short distance from the door. At least I assumed it was a chicken coop. Because now it was mostly splinters. As I stated, there were also a lot of dead chickens. They were littering the yard, lying still, leaving only the wind to be heard. I started keeping a chicken coop out here. The doctor said, something to do in my old age, and now something has done all this. I was sleeping in bed a couple nights ago when I heard something that sounded like the devil himself tearing through my yard. Did you get a good look at what did it? I asked. You kidding? It sounded like a bear was right outside my house. I wasn't going to go out there and deal with that. This seems like more a problem for animal control. Nolan said, See, I thought that, but then I noticed something. There's no blood. The doctor replied, What? I said, and I looked around the yard again and sure enough, he was right. Despite the volume of dead animals, there wasn't a drop of blood anywhere in the yard. The doctor walked over to a dead chicken and casually picked it up. He walked back over to us and parted some feathers near the chicken's neck to show its pink skin. Look, he said, there's just this bite mark. Me and Nolan looked and sure enough, he was right. There was a series of punctures along its neck, but otherwise the chicken was unharmed. It's the same with all of them. The doctor said, I don't know any animal that attacks like this and I got desperate for answers. That's what I called you. Nolan put his hand to his chin, seeming to think for a moment. Did any of your chickens survived it? Any at all? He said, Yes, actually, one of them slipped through the fence and then they came running back. Hold on a sec. The doctor walked back into the house. He emerged a few moments later, carrying a dog kennel. He held it up so we could see the chicken inside. It didn't look like it had witnessed some brutal attack on its fellow chicken kind, but then again, it was a chicken. It probably didn't have much of a short term memory. Great, because I'm going to need that chicken and kennel, Nolan said. What for? The doctor asked. Nolan turned to me and grinned. Who said a trap? It was beginning to fall as the doctor drove off into the distance. Nolan had convinced the doctor to leave the house for the night while me and him conducted our investigation. Nolan had given the doctor his contact information as collateral. So if we just wreck the place or stole anything, then the doctor could just call the cops on us. After the doctor had driven off, Nolan opened the trunk of his Lincoln Continental inside was a black suitcase. He took it out and then walked back into the house and I followed him. Nolan took a seat on the couch and I sat next to him. He popped open the briefcase and inside were several guns. Whoa, whoa, whoa, whoa, I said, scooting away from Nolan. Where'd you get those? Relax. Nolan said, these are just BB guns, really powered BB guns, but BB guns nonetheless. Nolan picked up two guns that looked like glocks, but with a way smaller muzzle.  He said, I can't get my hands on regular guns, but these are the next best thing. The BBs are silver, which is good since we're hunting a chupacabra. That's what you think did this. I thought those were just some urban legend. I said, Nolan turned to look at me with a bemused look. Right. Sorry, I said, but how can you be so sure? Well, let's take a look at the facts. Nolan said, the good doctor heard a large animal wreaking havoc in his own backyard. When he goes out there, all his chickens are dead, but there isn't a drop of blood anywhere. A chupacabra fits the bill perfectly. In fact, chupacabra translates to goat sucker because the first report indicated that it would simply drink the blood of livestock, including goats. Okay, I said, slowly catching on to what Nolan was really saying. So you think that because this thing is technically a vampire, silver will hurt it? Not just hurt it, my dear Watson. Nolan said as he pulled the slide back on his BB Glock, kidded it. The moon cast a pale glow over the country landscape as me and Nolan sat in some lawn chairs on the roof of the doctor's house. Below us, the sole surviving chicken, which we were using as bait and which Nolan had taken to calling Hoboken for some reason was clucking away. We were each holding a BB Glock in our hands, drinking some Dr. Peppers that the good doctor had left in his fridge. As we sat there sipping soda while waiting for an urban legend to show up to chow down on poor Hoboken, a question came to the front of my mind. How many of your cases end with you killing something? What are you talking about? Nolan asked, well, when we went to look at that basement, you seemed ready to frag whatever you found down there. And now here we are, not 24 hours later, waiting for some vampire dog to show up so we can waste it with airsoft guns. BB Gunch, Nolan corrected. He was silent for a moment. Look, you saw how the doctor was acting. He said, dude was modified to even be talking to us. Most people don't want to admit when something truly strange like this has happened. They'd rather just ignore it, move on with their lives or their jazz. But the thing is, ignoring something doesn't mean that thing will go away. If the doctor had contacted me, then this chupacabra would have moved on to the next poor guy's farm. So there's where I come in. I don't ignore the strange. I study, and if it's causing problems, I'd deal with it. Nolan gestured with his soda towards where Hoboken was clucking into the night. That's what I'm doing right now. What? We're both doing right now. I corrected. Nolan chuckled. Yeah, I guess I'm gonna have to get used to saying that, partner. I grinned. I raised my soda. I'll drink to that. Nolan smiled. Same. And with the tapping of two Dr. Pepper cans, the Nolan Beckett and Joseph Garrett Paranormal Detective Agency was born. As we sat there drinking our sodas for the occasion, and putting our phone numbers into each other's phones, because obviously you need to be able to contact your business partner, we're professionals after all, I realized something. Nolan, Hoboken stopped making noise. I said, Nolan turned to where the chicken should have been. The kennel had been torn open. Bits of plastic and metal were scattered around the yard. But there was no blood. Shoot, we missed it. I said, wait, Nolan said he rose from his lawn chair. I hear something. I got up from my lawn chair, holding my BB Glock in my hand. I looked around, but I couldn't see anything, at least not anything that looked like a vampire dog. Then there was a sound, a low gravelly growl that made my blood turn to ice. I looked out to where Hoboken's kennel had been. Two pure red eyes were staring back at me.  It was a little bigger than some wolves I'd seen at a nature preserve once. But this dog didn't have any fur. Its body looked emaciated, too, like it hadn't eaten in days. It reminded me of an animal with mange. And then it growled again. And then it started to run towards the house. Shoot it. Nolan yelled. He took aim and started firing BBs at it. The gun only made a small thunk thunk sound as the CO2 propelled the BBs. I saw the Tupacabra actually flinch as some made impact, but I was frozen solid. I couldn't even aim my gun at the thing. And then it jumped with the most strength I'd ever seen in an animal. It cleared the house and it was going to collide with me. Its muzzle was open, revealing its sharp teeth. And then my survival instincts kicked in and my body moved on its own. Without even thinking about it, I dropped the gun and grabbed the lawn chair, raising it as a shield between me and the beast. The Tupacabra collided with the lawn chair, knocking me flat onto my back. I held the chair with both hands as the thing tried to bite and claw its way through the scraps of vinyl and tubes of metal trying to get me. Over the gnashing and gnawing of the Tupacabra, I heard the thunk thunk of Nolan's gun. I saw the beast flinch as BBs embedded themselves into its side. The BBs cut through its flesh like a hot knife through butter and smoke began to curl out of the entry holes. The beast winced and I sensed an opportunity. Using all the strength I had, I pushed up with the lawn chair, pushing the beast off of me. I tossed the chair aside and quickly got back onto my feet, grabbing the glock I dropped. Nolan walked up to stand at my side. You alright? He asked. I'll live. I replied. Then we both turned to look at the monster. It stood on the roof in a position like it was ready to pounce at us.  Nolan asked me. Yeah. Yeah, I am. We both took aim and began to pelt the thing with BBs. The creature began to take steps backwards towards the edge of the roof as we unloaded our silver ammunition. Don't let up. Nolan yelled. Eventually, we pushed the thing to the edge of the roof. Its foot slipped as it tried to find more roof to back onto only to be greeted with empty air. The beast slipped and tumbled over the roof, yelping as it hit the ground. We heard a sharp crack. We stood on the edge of the roof and looked down. It looked like the thing had broken its neck on impact. Nolan sat down, feet hanging over the edge. I sat next to him. Welcome to the life of our detective partner. After the doctor returned in the morning, we explained the situation. The doctor did pay the rest of Nolan's fee since we had solved the case and dealt with the problem. But he dismissed the explanation of a chupacabra and insisted it was some kind of dog with a strange case of mange. Nolan didn't bother correcting him. And as the doctor laid out his explanation, I was reminded of what Nolan had said about people wanting to ignore the supernatural and strange. The power of denial can go a long way, I guess. We both drove back into the city from the Doc's house. I arrived at my house in the late afternoon. My parents started to ask me about the hangout and I informed them that I did indeed have some fun, but I really just wanted to nap right now. I walked upstairs and into my bedroom. I flopped onto my bed and I closed my eyes. The fact that I hadn't slept for a day hit me all at once and I felt myself drifting off into a wonderful abyss of unconsciousness. And then my phone proceeded to vibrate like mad. I dug it out of my pocket and looked at the notifications. Nolan was sending me text message after text message. I opened my messaging app and looked at what Nolan had sent. You ready to go partner? I chuckled. I guess this is the life of a detective. Well hello everyone, Detective Nolan Beckett here and I'd like to thank you for listening to my scary tale from Chronic Awesome. Make sure to check out his Reddit page for more of my updates. If you'd like to support a narrator of this tale, Lighthouse Horror, there is a page around in the description and remember, if your pet goes missing and there's no blood, it's probably a super camera.


The thing that I would focus on in order to for job creation is really what our industry and manufacturing and exporter sector is looking for. What they're looking for is training and it's a pull system. So right now we have what I would call a push system which is training people and then hoping they can find the jobs. Right now what we have is a load of companies that need expertise, they need the expertise right now and they need some form of internal training, training within industry. That helps to close the experience gap so I'll give you an example. Apprenticeship programs are really great but somebody coming out of apprenticeship program replacing somebody on a CNC machine that has 30 years experience doesn't cut it. The experience gap is so large that the individual coming into that machine cannot hope to replace the person with 30 years experience. For a lot of manufacturers it's not like the construction industry. If you have somebody that's come through a trade as a carpenter and can go into construction there's still an entry level job within construction knocking in two by fours or whatever. If I go to a specialized niche manufacturer say back to the CNC shop there isn't an entry level. So that's where the big gain can be had I believe is training within industry and setting up programs and helping to fund programs like that.


We got some advice from MrBeath. Just fail.  Period. Your first 10 are not gonna give views. I can very comfortably say that. Stop sitting there and thinking for months and months on end and just get to work and start uploading. All you need to do is make a hundred videos and improve something every time. Do that. And then on your hundred and first video, we'll start talking like maybe you can get some views but first hundred are gonna start. For the average person like us, just make a hundred videos, improve something each time and then talk to me on your hundred and first video. I agree with him, man. Artists always expect everything to go amazing out the gate. I look at it as a marathon of attrition. Who can stay in it long enough to earn the skills that make them better? Who can stay in it long enough to learn how to be entertaining? But that's probably gonna take you a couple of hundreds of videos to even get decent at it. Discouraging is that my sound? It's also kind of liberating, man.


in today's lecture we will be studying recurrence relations our starting point is discrete numeric functions which we have discussed in last two lectures suppose a is a discrete numeric function which is written as a equal to a sequence of real numbers which we denote in general by a 0 a 1 a 2 and so on then a n minus 1 a n and onward now sometimes what happens is that the entry a n of the discrete numeric function can be related in certain ways to the previous entries for example let us look at this relation a n equal to a n minus 1 plus a n minus 2 for n greater than or equal to 2 and a 0 equal to 0 and a 1 equal to 1 now we start from a 0 and of course it is given that a 0 is 0 then we come to the next entry that is a 1 which is also given as a 1 is 1 and then a 2 according to my rule over here is a 1 plus a 0 which is equal to 1 again after that a 3 which is equal to a 2 plus a 1 which is 1 plus 1 equal to 2 then we have a 4 which is equal to a 3 plus a 2 which is 2 plus 1 that is 3 then a 5 equal to a 4 plus a 3 this is 3 plus 2 equal to 5 and we have to proceed in this way thus we have a sequence whose first few terms are 0 1 1 then 2 then 3 then 5 and then of course 8 and so on now this is an example how a discrete numeric function can be generated recursively and we get very interesting sequence sequences for instance the sequence that we have just seen is the famous Fibonacci sequence the relationship that we get here is called a recurrence relation and the conditions that we write down as a 0 equal to 0 and a 1 equal to 1 are called initial conditions now we can have many more examples of recurrence relations for example we could have written a n equal to 3 a n minus 1 minus 2 times a n minus 2 so I write a n equal to 3 times a n minus 1 minus 2 times a n minus 2 and I start from n greater than or equal to 2 and for a 0 and a 1 I have to choose some initial conditions like this we can have recurrence relations like a n equal to n minus 1 a n minus 1 plus n minus 1 into a n minus 2 or we could have had a n equal to a 0 a n minus 1 plus a 1 a n minus 2 plus and so on up to some a n minus 1 into a 0 where the corresponding discrete numeric function is a 0 a 1 and all that so we see that at each term over here we have a product of two entries of the discrete numeric function then we could have had a n square equal to a n minus 1 square minus 1 so we see that there are many many different discrete numeric functions but the essence of sorry there are many many different recurrence relations but the essence is that given a discrete numeric function a 0 a 1 a n minus 1 a n I try to build up a relationship involving a n equal to the previous terms some f of a 0 up to a n minus 1 our goal in this topic is to find out expressions of a n purely in terms of n so we would like to find out a function of n explicitly written in terms of n which gives me the values of a n for n let us say greater than or equal to some number n 0 now in general this is a difficult problem and there is no general technique of handling this problem if somebody gives us a general recurrence relation but we can restrict the class of recurrence relations that we consider and build up some strategies or some general techniques of solving those recurrence relations so now our job is to find out a special class of recurrence relations and these will be called linear recurrence relations so I go to the next page linear recurrence relation alright so in case of a linear recurrence relation we will write a n in terms of some coefficients and the previous values of the discrete numeric function so I have c 1 a n minus 1 plus c 2 a n minus 2 and so on up to ck a n minus k plus fn here in general the coefficients ci's can be functions of n but we further reduce this class to a more restricted one called linear recurrence relations with constant coefficients linear recurrence relations with constant coefficients are those linear recurrence relations for which the ci's are constants so in this case ci's are constant for all i equal to 1 2 and so on up to k and now one would like to ask that what is this k is of course something less than n and it k signifies the situation that we look back up to some steps but we stop at after after a while so k is called the order of the recurrence relation k is called the order of the recurrence relation for example if k is 1 the recurrence relation will be a 1 a n equal to c 1 a n minus 1 if k is equal to 2 then the recurrence relation will be a n equal to c a c 1 a n minus 1 plus c 2 a n minus 2 if k is equal to 3 then the recurrence relation will be a n equal to c 1 a n minus 1 plus c 2 a n minus 2 plus c 3 a n minus 3 so the first one will be called first order recurrence relation first order the second one is second order the third one is third order and so on so we see that when we are looking at the first order linear recurrence relation then we are just looking one step back and a n is some constant times the previous entry to a n when I am looking at the second order recurrence relation linear recurrence relation let us say with constant coefficients then a n is nothing but a constant multiplied to a n minus 1 plus another constant multiplied to a n minus 2 and similarly for the third or the fourth and so on then we question what about this fn this fn can be identically 0 that means it is possible that fn is 0 for all n now if fn is 0 for all n then we get a n equal to c 1 a n minus 1 up to c k a n minus k this recurrence relation is said to be a linear recurrence relation which is homogeneous or homogeneous linear recurrence relation now it can be with or without constant coefficients if fn is not 0 then we have a non homogeneous linear recurrence relation so we have a classification of recurrence relations so we started with general recurrence relations which can be absolutely anything but we restricted ourselves to linear recurrence relations and a sub class of linear recurrence relations called linear recurrence relations with constant coefficients and within recurrence relations there are two different classes linear recurrence relations get split up into non homogeneous linear recurrence relations and homogeneous linear recurrence relations now what we are going to do is to further restrict ourselves so we will restrict ourselves to k equal to 2 so the class that we are going to consider now is linear recurrence relations with constant coefficients having order 2 first we will check order 2 and in the process we will automatically understand what to do with order 1 now if we have this then the recurrence relation will be of the type a n equal to c 1 n minus 1 plus c 2 a n minus 2 plus fn now I further restrict I make fn equal to 0 so I take the homogeneous case which gives me a n equal to c 1 a n minus 1 plus c 2 a n minus 2 my m is to find a way to write a n purely as a function of n equal to some g n and let us see how to do that so we have a recurrence relation of this type where c 1 and c 2 are constants and a is are values from some discrete numeric function we have to find that function what we do is that we take a so-called trial solution we say that let a n is equal to some capital A times r to the power n I do not know r but I know that a is a constant and r is also some values that we would like to find out we place this in the equation above to get a r to the power n and we transpose the elements of the right hand side to left hand side so I get c 1 a r to the power n minus 1 plus c 2 a r to the power n minus 2 equal to 0 and from this we get r to the power n minus c 1 r to the power n minus 1 plus c 2 r to the power n minus 2 the whole thing inside a parenthesis and into a equal to 0 it is very reasonable to assume that a not equal to 0 because is the constant a is equal to 0 then we have nothing to do we have the solution a n equal to 0 and of course that is a solution but we can do very little with that that solution so we write the equation as there is a mistake here it will be minus instead of plus so I replace by minus here and this is also minus so now I have minus and here also this is minus so minus c 2 r n minus 2 this is equal to 0 now again we can do something that is we can take the common factor r to the power n minus 2 out from this expression so I have r to the power n minus 2 equal to r square minus c 1 r minus c 2 equal to 0 now again it is very reasonable to assume that r is not equal to 0 because if r is equal to 0 again we have the 0 solution which is of course a solution in this case but it is it is useless so we have r square minus c 1 r minus c 2 equal to 0 we see that proceeding as above we have arrived at a degree 2 polynomial equation and we know how to solve that so we can write the solution as r equal to 1 by 2 into c 1 plus or minus root over c 1 square plus 4 c 2 at this point we have the well-known theory of solving quadratic polynomial equations in one variables and we know that r has 3 choices r can have two different values for which this quadratic equation is satisfied or we can get a single value that is c 1 by 2 which satisfies this quadratic equation or this expression inside the square root can be a negative one and therefore r may admit only two complex roots complex that is the equation will be admitting only two complex solutions in this lecture we will restrict ourselves to the case where one two distinct real values of r are available to only one real value of r is available which is the repeated root the repeated root of the equation or one can say the polynomial r square minus r 1 c 1 r minus c 2 repeated root so there are two cases that we will be studying in this lecture now the first case is where I have got two different solutions I call them r 1 and r 2 so I know that r 1 square minus c 1 r 1 minus c 2 is equal to 0 I also know that r 2 square minus c 1 r 2 plus minus c 2 equal to 0 so what I see is that I if I go back to the if I let us say if I build two different solutions one as r equal to some capital A times r 1 n and r equal to some let us write a 1 and let us write a 2 r 2 n then I can replace each of them in the recurrence relation which is given by a 1 minus c 1 a n minus 1 minus c 2 a n minus 2 if I replace just I should write this as a n as we have taken as a solution it is not r but this is a n so this is a n and this is also a n what I want to say is that the a n has two possibilities so then if I put a n as the first possibility of a n in the recurrence relation I will see that a 1 r 1 raise to the power n minus c 1 a 1 r 1 raise to the power n minus 1 minus c 2 a 1 r 1 raise to the power n minus 2 which is equal to a 1 r 1 raise to the power n minus 2 r 1 square minus c 1 r 1 minus c 2 and this is of course equal to 0 so this of course satisfies the recurrence relation given by this and similarly the other solution also satisfies the recurrence relation but what is interesting is that if we add these two solutions and get something like this a 1 r 1 raise to the power n plus a 2 r 2 raise to the power n let us call this a n now if I replace this in my in the left hand side of the recurrence relation which is essentially this then I get a 1 r 1 raise to the power n plus a 2 r 2 raise to the power n minus c 1 times a 1 r 1 raise to the power n minus 1 plus a 2 r 2 raise to the power n minus 1 minus c 2 a 1 r 1 raise to the power n minus 2 plus a 2 r 2 raise to the power n minus 2 and rearranging the terms I will get the terms that we have already got that is a 1 r 1 raise to the power n minus n minus c 1 a 1 r 1 raise to the power n minus 1 minus c 2 a 1 r 1 raise to the power n minus 2 plus a 2 r 2 raise to the power n minus c 1 a 2 r 2 raise to the power n minus 1 minus c 2 a 2 r 2 raise to the power n minus 2 now this is of course 0 first first term is 0 we have already seen over here second term is also 0 that we can check because after all this is a solution so 0 plus 0 is equal to 0 thus we see that we can construct essentially an infinite number of solution by taking linear combinations of r1 raise to the power n and r2 raise to the power n I am essentially free to take any constant values for a1 and a2 thus we have obtained several solutions of the linear recurrence relation we started with but what is most surprising is that advanced theory of recurrence relations tells us that for the reconciliation under consideration these are essentially all the solutions of that one therefore the job that remains for us is to put the initial conditions and obtain the values of the constants which will give us the particular recurrence relations that we are looking at we will check the first recurrence relation that we built up which is corresponding to Fibonacci sequence the refer recurrence relation is an minus an minus 1 minus an minus 2 equal to 0 for all n greater than or equal to 2 along with the initial condition a0 equal to 0 and a1 equal to 1 now let us take the trial solution as a n equal to a constant times r to the power n and if I put this expression in the right hand left hand side I will get a r to the power n minus a r to the power n minus 1 minus a r to the power n minus 2 equal to 0 and then if I take the common factors of all the terms then I get a into r to the power n minus 2 multiplied by r square minus r minus 1 equal to 0 and we know that this means that I have to solve the quadratic equation given by r square minus r minus 1 equal to 0 and we can of course do that by writing the general form and which is r equal to 1 half of 1 plus or minus 1 plus 4 which is equal to 1 plus or minus root 5 by 2 therefore the general equation for the general solution of this recurrence elation is of the form an equal to some constant a1 times 1 plus root 5 divided by 2 raise to the power n plus a2 1 minus root 5 divided by 2 raise to the power n this is for n greater than or equal to 2 and for a 0 I know it is 0 and for a1 it is 1 now what I do here is that I try to fit the expression that I have got over here from n equal to 0 onward so I force a 0 equal to 0 and force the this expression in the right hand side to get a1 plus a2 and then 1 equal to a1 times 1 plus root 5 by 2 plus a2 times 1 minus root 5 by 2 that is all and so I have generated two equations suppose I am able to solve them and get the values of a1 a2 then I can plug in the values of a1 a2 in this equation to get a solution for the Fibonacci sequence reconciliation so what I do here is that from 1 we have a2 equal to minus of a1 substituting into we have a1 equal to 1 plus root 5 divided by 2 minus 1 minus root 5 divided by 2 which should be which should be equal to 1 and this is same as a1 times 2 dividing 2 root 5 equal to 1 2 gets cancelled so I have a1 equal to 1 by root 5 and a2 equal to minus 1 by root 5 therefore we have arrived at the solution of the recurrence relation that we are considering here we will say a n equal to 1 by root 5 into 1 plus root 5 by 2 raise to the power n minus 1 by root 5 1 minus root 5 by 2 raise to the power n for all n greater than or equal to 0 thus in today's lecture we have introduced the idea of recurrence relation then we have introduced linear recurrence relations and restricted though that to the class of constant coefficients we have introduced the classification homogeneous and non homogeneous recurrence relations linear recurrence relations and lastly we have considered a homogeneous linear recurrence relation with constant coefficients which corresponds to Fibonacci sequence and completely solved that relation this is all for today's lecture thank you


Welcome back to our lecture series Math 4230, Abstract Outdoor 2 for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misseldine. Lecture 16 is going to focus on the idea of factorization, and in particular, we're going to build up to the idea of a unique factorization domain for which we're going to define into the next video, part two of lecture 16 here. Before defining what a unique factorization domain is, we need to actually clarify certain ideas about factorization in commutative rings with unity. Because after all, if we go back to some of our earliest learnings of mathematics beyond basic addition and subtraction and such, some of the first conceptual ideas we learn about the integers, whole numbers as they might have been called, are ideas of factorization to visibility. Where this isn't just like rote calculations where we learn how to add and we can memorize that, or we memorize multiplication tables or things like that. Factorization was one of these first hard problems, and we sometimes draw factorization trees and things like that to find prime factorizations. If we have two different numbers, we can factor them and try to find their greatest common factor or least common multiples, and there's some important applications to understanding these, but factorization and the related concept of visibility are some of the most fundamental yet first conceptual ideas we learn in our mathematics educations. Now, in this lecture 16, and in the next coming lectures as well, we want to start developing the notion of factorization and visibility for the general category of rings, or more specifically, we're gonna focus on commutative rings with unity, and even more specific than that, we're really gonna be talking about integral domains. Because it turns out that outside of an integral domain, factorization can be very strange. It can be a very strange thing. It doesn't exactly behave the way we're used to, at least not from our primary education whatsoever. But even inside an integral domain, things can get a little bit weird as we'll see some examples of that. And so that's what's gonna lead to the idea of a unique factorization domain, for which those are exactly the integral domains where the rules of factorization work the way that we're used to. In other words, they work the way that they do for whole numbers, which is sort of coming up ultimate goal. So I should mention that as we go forward with lecture 16 into the future, when we talk about a ring, that to us is gonna mean a commutative ring with unity. I'll usually try to say it, but it is somewhat of a mouthful. So if I just say ring for short, I want you to assume that in these lectures about factorization, which of course they coincide with chapter 18 in Tom Judson's abstract algebra textbook, this ring means a commutative ring with unity. Because if we don't have commutivity or if we don't have unity, rules of factorization can be very, very alien. And we want to look at those rings that factor the same way that the integers do. That is leading up to this unique factorization domain. So there's some important terms we have to talk about before we can define what unique factorization even means. So imagine we have a ring with unity, call it R, and take two elements A and B that are inside that ring. What does it mean for A to divide B? Well, we say that A divides B if there's some other element C inside the ring such that AC is equal to B. So A divides B if A is a factor or divisor B because there's some product of B involving the number A. So just like we do when we talk about integers, we're gonna use the symbol A dash line or whatever you call that. You're gonna use a line here, A divides B. That's how we read that. And I should mention that this divisibility symbol is gonna form what we call in mathematics a preorder as a relation. A preorder is, it's kind of like the intersection between partial orders and equivalence relations. A preorder is gonna be something that's first of all, reflexive. And so why is this reflexive? Well, it's reflexive because you have unity, right? In particular, A divides A because A can be factored as A times one. So that's one of the reasons why we need to have unity inside of our rings in this situation. But we also have that it's gonna be transitive. Transitive. So if A divides B and B divides C, then we can infer that A divides C. And so why is that? Well, if A divides B, that means there's some AD that's equal to B. If B divides C, that means there's some BE that equals C. And so we can put those together because you can substitute this into there. So we end up with AD times E equals C. But by associativity, you can redo this and you get A times DE is equal to C. So it's gonna be a transitive relationship. Is it a partial order? Is it an anti-symmetric relationship? Well, it kind of depends. I mean, if you look at the de-factorization of positive integers, that does give you a partial order. But if you look at the factorization of integers, then you don't get a partial order because the anti-symmetric property doesn't happen. In the ring of integers, you can have two that and two divides negative two, since negative two is two times negative one. But you also have the negative two divides two because in that situation, two can be factorized negative two times negative one in that situation. And so in general, this divisibility relation on a ring doesn't give you a partial order because the anti-symmetric property doesn't always hold. So like I said earlier, pre-orders, that's actually the intersection between equivalence relations and partial orders. Basically, you're dropping the symmetric or the anti-symmetric. You're just taking reflexive and the transitive relations. That's why it's called a pre-order. It's getting ready to be a partial order, but you don't necessarily have anti-symmetry. Equivalence relationships are also pre-orders because they have symmetry. That's to say, they have the reflexive and transitive properties. And so when it comes to divisibility, if it's not a partial order, it's because of this issue with anti-symmetry. You could have these numbers that divide each other. Now, because of this phenomenon, it actually turns out we should give it a name. If you have two elements in a ring, call it R and S, then we say that R is an associate of S. If there exists a unit U inside the ring such that R equals US, okay? So if R, notice in the situation, we have that S divides R. But since U is a unit, we can actually rewrite this as U inverse R is equal to S. So we see that R divides S. So R divides S and S divides R, they divide each other. That's exactly what an associate is, although this will be our formal definition. Two elements are associates of each other if there is a unit which multiplied by one gives the other. Now, speaking of pre-orders here, the relationship that R is an associate to S is an example of an equivalence relation. An element is an associate to itself because like we said earlier, R equals R times one. It's transitive because being an associate means you divide each other, right? If R is an associate of S, excuse me, if S is the way I phrase it here, if S is an associate of R, that means R divides S, okay? But we also have the other way around, you know? If S divides its associate T, then you're gonna get, in fact, that R divides T. R divides T. So because of the visibility relationship, you're gonna get transitivity for associates as well. But now this is where symmetry comes into play like we said a moment ago, right? Because the unit can come over and become its inverse, so you get U inverse R equals S, you're actually gonna get symmetry as well. So if R is an associate of S, S is an associate of R. So being an associate forms an equivalence relation like we see over here, you have these association classes inside of a ring with respect to these factorizations. And the association classes are the obstruction from divisibility being a partial or if you're not, you're gonna be anti-symmetric only if no elements have different associates, if you're only associates to each other. Like if you look at the natural numbers, they don't have any associates because their only unit is one. If your only unit is one, then of course you have no other associates. But like in the integers, you do get the unit negative one, which is really typical. And so in general, we're not gonna get a partial over there, but this studying of associates is really important. When it comes to factorization, there's really no reason to distinguish between associates. So when we talked about two before, right? You know, two can factor as negative two times negative one. This factorization is kind of trivial. It's like, oh, you just replaced two with an associate. If one of your factors is a unit, it kind of doesn't matter in terms of the factorization. And so that's then gonna lead to the idea of an irreducible element inside of a ring. So if M is an element of commutative ring with unity, we say that M is reducible if every factorization of M involves a unit, which is equivalent to saying that every factorization of M involves an associate of N. So in other words, if M equals AB, so if you have a factorization of M, irreducible means that either A was a unit or B was a unit. Remember the set R star here represents the units of the ring. Related to the idea of irreducible elements is the idea of a reducible element. A reducible element means it has a factorization where the two elements involved are not units. And so in some regard, a factorization of reducible element then gets you something smaller, right? We were factoring 12 before. 12 is a reducible element inside of the integers because we can factor this four times three. Four is likewise reducible because we can factor this two times two. Three, on the other hand, is an irreducible element because every factorization of three involves either one or negative one. In other words, it involves the three itself or it's associate negative three. And the same thing can be said for two. Two and three are gonna be these irreducible elements because there's no factorization of them that gets you something simpler than what you already had. Every factorization involves a unit or an associate. Now, don't we usually call two and three prime numbers? Why are we calling them irreducibles? Why don't we just call this a prime element? Well, it turns out in ring theory, we have a notion of a prime element and it's actually defined differently than irreducibility. In primary school, when we learn about a prime, what we actually defined is an irreducible number, not a prime number, which I confess irreducible numbers a little bit more of a mouthful for little children. But nonetheless, when we define prime numbers in primary school, we're like, oh, three is a prime because every factorization of three is one times three. We're ignoring negative numbers because students often haven't learned about negative numbers. But because the negatives are just associates to positive, that's okay, we can ignore them because the inclusion of negative numbers, the inclusion of the negative unit, negative one, doesn't really change the factorization, it's irrelevant. And so we define prime numbers actually to be irreducible numbers, okay? So what's a prime number then? What's the real definition of a prime? So we say that P is a prime element of a commutative ring with unity R if every product, if any product A, B and R has the property that P divides it, then either P divided A or P divided B. So say that again, we say that a number, or what, a number, what's a number, right? A number is just an element of a ring, I guess, right? You know, if we have some element of a ring that divides a product, that element is prime if for whenever it divides the product, it divides either the first factor or the second factor. Now, this statement right here is a resemblance of what's commonly referred to in number theory as Euclid's lemma. Euclid's lemma says that if a prime number, speaking of integers right now, if a prime number divides a product, then it divides one of the factors in the product. So we're actually taking Euclid's lemma to be the definition of a prime in a general ring, right? If an element's prime is whenever it divides a product, it must divide one of the factors. So if we were to use the language of ring theory, which we now have on the screen, we would actually say that Euclid's lemma is the following, irreducible numbers or prime numbers, at least in the ring of integers. We're gonna see that in general, that's not always the case. Now, in the case of an integral domain, it is true that primes are always irreducible, but in a later video, I'll actually provide you an example of an irreducible element in an integral domain that's not a prime element. That's not the case for the ring of integers, which is why in primary school, we can abuse the term irreducible versus prime, but in general domains, we have to be careful we distinguish between these objects, but in integral domain, primes are always irreducibles. Prime being a prime implies you're irreducible for the following reason. Imagine we have a factorization of a prime element, we're assuming P is prime here. So to be irreducible, we have to prove that either A is a unit or B is a unit, okay? Well, it's always true that P divides itself, right? Because you can always factor an element as itself times one. So P does divide P times one, okay? We also know that P divides AB, right? Because I mean, it's equal to AB, that's what we're trying to say right here. Since P divides itself and itself is equal to AB, we know that P divides AB. Why is that important? Well, that's part of the definition of being prime. Since P can be factored as A times B and P divides itself, P divides AB. Because it's prime, then by the Euclid property, the definition of a prime, it must divide A or it must divide B. We are in a commutative ring, so it really doesn't make any difference which factor we use, because the first one could be the second one doesn't matter. Without the Lawson General, we may assume that P divides the factor A. Now, if P divides A, that means there exists some element C inside of the integral domain D, such that PC is equal to A. Well, again, that's what divisibility means. That's the definition. And therefore, when we come back to the original factorization, P is equal to AB. Since P divided A, that means that we can replace A with PC times B. We're in an associative ring here, so we actually end up with this thing right here. This is P times CB, like so. And so notice the factorization. We have P is equal to P times CB. Wait a second, we're in an integral domain, right? We haven't used that fact yet. Well, because we have a P times one is equal to a P times CB, in integral domains, we can cancel, we left cancel, and so we end up with CB is equal to one. That implies that B was a unit. Remember, we said P divided A. Well, the one that P didn't divide actually was a unit. And so since the factorization of P, this was an arbitrary factorization, always involved a unit, that implies that P is an irreducible element. So in integral domains, primes are always irreducible, but be careful. In the future, we will see an example of an element, but at the end of this lecture, in fact, we'll see an example of an element which is irreducible, but not prime.


Today, Ralph joins us from Pennsylvania. He's Freedom Flyer number 214. He spent 14 months in captivity up until 1973. I tried to eject, but we didn't do it. The 560th Flying Train Squadron hosts the Freedom Flyer Union every year here in March at Randolph Air Force Base. We host POWs who've returned from captivity in North Vietnam during the Vietnam War and returned them back to flying service. Back in their early 70s, they returned to flying status on active duty. And now today is mostly a celebration of their service and bring their friends and families to put them back up to in 238, put them in a flight suit and return to flying again with their Air Force friends. The amazing thing is just walking through the hallways and seeing the history of their involvement with the POWs. The significance of the POW cause and the missing inaction cause, the memorabilia, the artwork here behind me is really a testament to the commitment these folks have made on an ongoing basis now for a decade or more to ensure the POWs are not forgotten. So it's really, really impressive what they've done. When you go on an active duty base and realize that even the senior officers are younger than your kids, but it's still fun. The energy is high. There's a commitment to service, commitment to their community. And that's something that I've seen off and on for the past 50 years, even after my time on active duty. So it still is a warming environment to go on a military base. It will be a great memory for me, but it's also important for me to come back and see the message and the history and the commitment of this squad of 560s especially, the commitment they've made over decades to ensure that POWs are not forgotten, that we get this one final mission. And recognition for what we've done is really impressive and greatly appreciated.


Mr. District Attorney, starring David Bryant, Mr. District Attorney, champion of the people, defender of truth, guardian of our fundamental rights to life, liberty, and the pursuit of happiness. And it shall be my duty as District Attorney, not only to prosecute to the limit of the law all persons accused of crime perpetrated within this county, but to defend with equal vigor the rights and privileges of all its citizens. This is David Bryant. In a moment we'll bring you another case from the files of Mr. District Attorney. To purse the word from our sponsor, David Bryant as Paul Garrett, Mr. District Attorney. To the untrained observer, a crime and an accident may sometimes be indistinguishable, one from the other. But a District Attorney must weigh each possibility in the scales of science and circumstances before drawing his conclusions. You take this case. It started on a farm near County Line at 2 a.m. As a man awakened by a wave of nausea, staggers weakly from his bed. May, where are you? What's the matter? Why are you on a bed? Oh, I feel so sick. You're kind of funny myself. I was just filling a hot water bottle for my stomach. What can be the matter with me? Honey, why does this sheet... You better get back and lie down. Stay at my feet without holding on to something. You're all perspired.  Must have come down with the flu or something. Their kid brother acted kind of funny before he went to bed. I heard him get up about 11 o'clock, complaining about a stomach ache. Yeah, we better call the doctor. Here he is here. Go throw an extra blanket over him. Harley, he's 20 years old. You can get a blanket if he needs it. Come on. We better get back to bed ourselves. Let me take you a minute to fix it, Bob. All right, you come on now. We'll call the doctor in the morning. Ah, he never felt so sick. He's very safe. He feels so funny. Bob, he's so cold. He doesn't even breathe. No, no, no. Don't get yourself excited. I'll wake him up. Walter, wake up. I'm going to call the doctor right away. No, but it's a hell of a problem. From each bottle for analysis. Anything? No. That'll be poison though. Three bodies? No sign of violence. No weapons. Longer the medical examiner, figure they'd been dead. Also, some time night before last. Long time. Did the lab men check the garbage cans and the refuse containers? Yeah. No empty bottles of any kind. Just a few cans, a jar, vegetable peelings, you know, things like that. Well, something killed him. I'm going to have another look around downstairs. He's taking it to town. You know, about three hours. The autopsy report should be ready pretty soon, then. Unless the doc runs into trouble. Yeah, well, I figured he wanted it as soon as possible, so I asked him to send it out with this bill. It was as soon as possible. Good. I think that the car outside now might be hurt. Oh, that's a man's knock. Come in. Which one of you is Gary? I am. Who are you? My name is Sid Mac. I own half this farm. I was Farragas' partner. I just want to find out one thing. I want to know who did this. If we knew that, we wouldn't be here trying to find out, Mr. Mac. I just want to make sure that whoever did it doesn't get away with it. You can post a $5,000 reward for the killer in my name. Make it ten. Make it anything I got, but get him. We don't need rewards for that, Mr. What we need is information. You said you own half this farm, Mr. Mac? Yes. And you haven't been here since the night before last? I don't live on the place. It was just an investment for me. I said Farragas, get up here. I got a big hardware store over in Emmettville. I live there. I see. Do you know anybody who might have had any reason for wanting Farragas' family out of the way? Out of the way like this? I don't have to be a madman. The squad car just turned into the road chief. That's Miss Miller. Looks like Simmons is driving. We've been waiting for an autopsy report, Mr. Mac. It may tell us which way to move. I hope so. Dr. Miller, is autopsy report, Mr. Garrett? Good. Could he establish the poisoning agent? There wasn't any. There wasn't any? You mean he couldn't make an analysis? No, sir. There just wasn't any deliberate introduction of a poison. All three deaths were accidental. Let me see that report. Doc must have missed something this time, chief. No, no. He didn't miss anything, Harrington. The deaths were caused by botulism. Botulism? What's that? The result of improper home canning of food. The report shows that the Farragas made their last meal of green beans, potatoes, and canned sausage meat. The bacteria was unmistakable. It was the home canned sausage meat. Well, I guess in a way we can be thankful, chief. It's nice to know it wasn't litter. In another way, I almost wish it were. Mr. Garrett. No, don't misunderstand me. I'm afraid we've got a greater and a much more dangerous job on our hands than just finding a killer. I don't follow you, chief. We've been all over the house, Harrington. The kitchen, the pantry, everywhere. There's nothing in the house that's home canned and no equipment for home canning. Hey, that's right. All we found was one empty jar of the kitchen grain board. Farm women do pass out samples of their home canning to friends and neighbors. Mr. Garrett, that jar might have been a gift.  What you're trying to say is that somebody around here might have a pantry full of poison and not know it. That's exactly right, Mr. Mack. The Farragas are dead. But dozens of others may die the same way unless we find out where that sausage meat came from and fast. Harrington kept it all available local police on this with a couple of squads sent out from the city. I want the sheriff and all his available men part, too. Right. Send them out for a 20-mile radius. I want direct contact with anybody in the area who can't be reached by phone. See if anybody they can reach give the Farragas a jar of sausage meat. Get right on it. Have it set up in 20 minutes. Ms. Miller, you'll be in charge of the phone campaign, special operators from the phone company, all that they can spare, all the things in this area that be called. So where will you be? Back at the office. Apparently, we don't get any more calls from the hospital or the morgue before we find the source of that jar. All right, you. Oh, that's an elevator. Hold it, Pete. I don't think we miss more than a dozen people in the area you mapped out. Now, we know most of those are away on business or vacations. Yeah, I know. Five days. If only somebody come forward and admit that they came to stop the Farragas safe. We know we were safe. They may be afraid of being held responsible for the death. Oh, with the newspaper, the TV and the radio campaigns? Tell them about the inquest how the death is acts of devil. That's the same. People scare. This is something to wonder about. Oh, here we are. Thanks, Pete. Maybe one of the squads will go up to the floor with this fellow. What time is it? Almost 10 p.m. Sorry to keep you so late again, Ms. Miller. Any news? I'm afraid not. Everybody's checked in. The reports are on you, guess. Oh, thanks. You'd better go home. All right. Well, there's a Mr. Carruthers in your office. He's been waiting to see you since about 6 o'clock. 6 o'clock? That's a long wait in any league. You know who he is? No, or what he wants? He wouldn't tell me. He said it's confidential. All right. I'll see him. You run along. Good night. Good night, Harry. Good night, Ms. Miller. Well, let's have a look at the reports and see what the very patient Mr. Carruthers wants. Excuse me. I almost dozed off. No, it's all right. I'm Paul Garrett. This is my assistant, Mr. Heinten. All right. Anything we can help you with? Yes. Yes, I flew down from Hartford to see you, Mr. Garrett. I'm an investigator for the Eastern Infrontier Insurance Company. Mr. Garrett, we understand that you are still investigating deaths of Robert and Mae Farragut and her brother, Walter McMahon. Well, we're trying to locate the source of the food that killed him, if that's what you mean. Then this isn't a criminal investigation. Didn't you get a report on the in-class verdict? Yes, sir. You can't blame this for being interested when a beneficiary submits a claim for $30,000. $30,000? Who's a beneficiary? Somebody who had a half interest in the Farragut farm. A man named Sid Mack. No wonder he could afford a war for a $10,000 reward sheet. Yeah. How long ago were those policies written, Mr. Carruthers? I have the dates right here. Just a little over a year ago when Mack bought his interest in the farm. That's the main reason my company wanted to be certain about your investigation, Mr. Garrett. It's a routine matter, of course, for partners to ensure each other, but this involves Farragut's entire family. Yes, it is. However, since there's no criminal investigation, you have to honor Mr. Mack's claim. Thank you for your time, Mr. Garrett. Just a moment, Mr. Carruthers. I suggest you don't recommend payment of that claim just yet. But, Mr. Garrett, you just said there's no criminal investigation. There wasn't a minute ago, but there is now. This is David Bryan. Before we continue with Mr. District Attorney in the case of the canned death, here is an important message from our sponsor. Back to David Bryan, starring as Paul Garrett. Mr. District Attorney. Connor had listed as accidental the death of three persons, all victims of botulism, a deadly food bacteria resulting from improper home mechanics. In trying to locate the source of the lethal food to prevent other fatalities, we came into contact with an insurance investigator who disclosed that the three dead people had been heavily insured and that the beneficiary was Sid Mack. Harrington and I got a search warrant for Mack's house. Mack had already left for his hardware store, but we were admitted by a hired girl who disappeared as we were making our search. There was nothing in the kitchen, Chief. There was nothing in the pantry either, Harrington. Let's try to sell it. Well, scooted enough, Harrington, Chief. I'll be gone to Mack's hardware store to tell him what we're doing. Yeah, he'll have kept it until we finish. Doesn't make any difference. He knows sooner or later. And if there's anything here, he won't be able to stop us from finding it. We're not going to find anything, Chief.   His whole job is too clever. No crime is perfect, though. If there was a crime, there's been a slip someplace. Let's move those crates. Okay, Chief. Try the barrel. Must be Mack. You down there, Carrick? That's right, Mr. Mack. You know so much about the law. Maybe you're going to need more of the search warrant. I had a call from the insurance man named Carrotta this morning.  That's why we're here. I've got a legitimate insurance claim, but you stop payment of it. It'll be paid in due time, if it should be paid. Yeah. Let me tell you something, Garrick. How did the farm, before we found out what killed the parishes, I offered to put up everything I have as a reward. Didn't I? Yes, Mack. You did. I'm glad you mentioned that, Mack. Because it just brought something to my mind. Something that's been trying to register ever since you made that offer. What? How long have you been on the hardware business? Look, eleven years. Why? Because when we told you the Farrakets died from botulism, from food that wasn't canned properly, we almost had to draw you a blueprint. You didn't seem to know anything about it. How should I know anything about it? Don't you sell canning equipment at the hardware store? All right. We can ride over to the store, have a look at your stock. Do you sell it or don't you? All right. I sell canning equipment. Any hardware store does. I don't even know why I'm bothering to talk to you. You've got your warrant. Go ahead and search. But you're not going to find anything here. No canning equipment and no canned sausage meat. So go ahead. Search your heads off. He knows something about those debt sheets. He practically told us right away. Yes, he's pretty sure of himself. He hasn't got a case and he knows it. He could have brought canning equipment home from the store then dished it when he was finished. You'd need more than just equipment behind it. What? Pork, cognate. Quite a bit of it too. Mochilism bacteria isn't easy to cultivate. Even deliberately. You could probably take quite a few tries before a poisonous batch developed. I don't know. We can check butcher shops. And the farms. He might have had one butcher to some farm around here. Yeah. Well, it means we've got to comb this whole section with the county again. Yeah, they'd do that or... Or what? Or let a man get away with a triple murder. I'm sorry, Chief. I didn't mean it the way it sounded. Let's go. You might have given me some kind of a lead though. Like what? Well, it seems there's a place we missed when we were making our first check around here. It's not really a farm. Well, what is it then? It's a place over Dake Road, about a half mile north of here. Far as this says, there's an old woman living in a shack back there with her grandson. Her name is Annie Hibbs. I wanted for us to say about her. She had a thing hard to remember the name Annie Hibbs, Chief. They usually call her by a nickname.  She raises hogs. That's our next stop. Far as this says, she's kind of strange. You know, her grandson is feeble-minded. They had him at the state asylum for a while, but he was harmless, so they let him go. And why are you looking so worried? Well, Hogs aren't the only animals up there, Chief. Far as this says, they've got the meanest dog in the state. And don't they keep him tired? Yeah, but he chews loose. And he hates everybody but the old lady and her grandson. Well, we've got to see those people. That's going to take more than a mean dog to stop us. Think then, too. Look, sir, there's a man. Look. Hey, hey, mister, wait a minute.  That must be the grandson. Now, let him go for now. Why did you stare at my grandson? What do you want, sir? Well, we'll tell you in a minute, but there's something else we'd like to know first. Where's your dog? That's him. Very there. The devil come for him. He's dead. And Luke cries for him. He's afraid at night without the dog. Luke must be the grandson, Chief. Yeah. We just came up here to see if you have any home canned sausage meat we might buy. No. I don't know nothing about canning. I see. Then you never gave any canned sausage meat to a family named Farragut. You know who they were, don't you? Their farm is less than a mile from here. Yes. I know. Then you must also know they're dead. Not dead. Grand glory. If you never gave anything to the Farraguts, did you ever give or sell, you know, any of the canned sauces to a man named Sid Mac? Or even a live hog? Well, did you any? You've got to write to sell what you own. I don't know nobody by that name. He owns the hardware store in Emmettville. That's the only town near here. I never sold him nothing. He'd never come up here. Oh, tell us that he never... It's all right, Harrington. Thank you, Annie. That's all we wanted to know for now.  When you come back to that place again! You'll never get your culture back off with living. You ain't taking Luke back to that place. You ain't taking Luke back to that place. She means the asylum, I think. Yes. Get over behind that brush. It's hard to tell with anybody like that. But she seemed to tighten up when you mentioned Sid Mac. Yes, he's been up here all right. Harrington, I've got a hunch. You stay here until I send the lad crew out. What for? I want to know what killed Annie's dog. Oh, yes, he's here. No words for Harrington yet? No, sir. Radio division tried again about 45 minutes ago and still can't reach him. Or the lab car. That's not where that dog was buried. It was close enough to the road for them to hear the radio calls. I've been out of my car. I'll have to go. Chief, we've got... The whole case is cracked wide open. Oh, what happened? Botulism killed the dog, all right. We caught Annie's grandson. When we carved him down, he told us the dog had died. After Sid Mac had bought a jar of meat up and up the feeder. And Sid is mine. And we still haven't got him cornered. Why not? There's Luke's background on the asylum and a smart defense attorney in court scaring him and confusing him. His story wouldn't hold up. Not even if we could produce a few more jars of the contaminated batch with Mac's fingerprints all over them? You've got them? More than 20 jars. Luke showed us where Mac buried them. About 100 jars from the dog's grave. So that's why radio division couldn't reach you? That's right. That's why. What time is it, Miss Miller? 10 past four. Good. Come on, Harrington. We can make Emmettville very nicely before Sid Mac closes his hardware store. Kind of a bright idea if you've got this term. An idea that's going to put you for murder, Mr. Are you kidding? You can drop that smile, Mac. We've got the whole story from Luke Hibbs and his grandmother. The words of a lunatic. Doesn't the law say something about a reasonable doubt, Mr. Garrison? Not when we've got some buried samples of your canning with your fingerprints all over the jars. Just yours. So, like it once said, I sell canning equipment and if I sold those jars, I handled them. And my prints were on them. I think my story is good on that point. I don't. Because those prints were put on the jars after they were filled and the canning completed, any prints that were on them before that would have been steamed off in the process of canning and proving that to a jury won't be hard. Don't you raise that core hammer, Mac. Don't make me put a bullet in you because, frankly, Mr. I'm tempted. Wait a minute. Don't let him talk like that, Mr. Garrick. I grabbed it. I'm not resistant. All right. Just keep it that way. Let's go. You've got to do all wrong, I tell you. It was all an act to save it for the jury. You'd better lock up. You won't be coming back. What can I do? What can I do to square this?  Go on. Lock it. Brian again. I hope you've enjoyed this case from the file of Mr. District Attorney. I'll be back in just a moment after this message from our sponsor. Mr. District Attorney, David Bryan, with a word about the program you have just heard. David Mack was tried and convicted on a charge of murder in the first degree. The death sentence was mandatory when the jury failed to recommend clemency. And now this is David Bryan inviting you to join us when we present our next case based on the facts of crime from the files of Mr. District Attorney. Mr. District Attorney was originated by a Philip Dates law.