This is stock number 20 J220. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and use Jeep and Jeep Grand Cherokee headquarters. Today we are checking out this brand-new 2020 Jeep Grand Cherokee Overland High Altitude Edition. This vehicle has the award-winning 3.6 liter Pentastar V6 motor. It is in slate blue metallic. We shoot all of our videos in 1080p 60 frames per second so if you have HD capabilities on your computer tablet or smartphone device turn them on right now because it is definitely your best way to check out the styling and looks and color of the vehicle before seeing it in person. If you want to check out all the photos on this Jeep in the upper right hand part of your screen is a link right to our website click then check us out there. This one's gonna come with the Prowley Scorpion 26550R20 tires comes with the granite crystal painted alloy rims part of that high altitude package and that's kind of a highway terrain tread pattern on there. Slate blue metallic really a good-looking color they came out with this a couple years ago as the HID headlamps the LED running lights the LED fog lamps and running lights down there front bumper parking sensors it also has the adaptive cruise control take note of the granite crystal bumper inserts around the bezels there and on the seven-slotted grille here the inserts on there also the Jeep logo is that granite crystal color as well get the blacked-out Cherokee lettering outlined in the granite crystal as well as when as the enter and go system we're gonna take a look at the original window sticker here feel free to pause this I'm not gonna go through all the options but we'll just check out the basics here. Slate blue is a 245 option the high altitude package which is a 2bs is a 2745 package and it shows you everything that you get with that high altitude package including the protect to stuff with the adaptive cruise control and parallel park assist and lane departure and all that good stuff you also get protect one standard which is the backup camera parking sensors and blind spot cross-path detection total MSRP is 54 180 highway 25 miles per gallon 18 city 21 average and my favorite part about the grand Cherokees is the safety rating up and down five and four stars five star overall very safe vehicle that you're gonna want to put your family in and they really have done well with the grand Cherokees you know on this particular body style and because it's been around a little while this body style definitely has all the kinks worked out you know not gonna run into too many problems with you know this particular Jeep Jeep it does have the towing package receiver hitch four pin and seven pin wiring it has the power liftgate the granite crystal Jeep lettering on the back there is the cover for that receiver hitch it's got two little pop pins on springs there factory subwoofer in here you also get a full-size spare tire those seats do fold on for extra storage it does have these sunshade as well which is a really nice feature if you got groceries or something you want to keep them cool out of the sun you know it's also a kind of a safety device as well you know if you've got expensive stuff computers whatnot put them under here and then it's kind of out of sight out of mind type deal and nobody's gonna break into your car to steal something they can't see anyways this power liftgate powers down like that press it beeps at you three times comes on down actually beeps a total of four speaking of four you got the four by four and the taillight bezels grayed out as well and coming down the driver's side here it does have the capitalist fuel fill four wheel disc brakes on the grand Cherokees of course and like I said the entering-go system you do get that blind spot monitoring in the mirror they are heated and you get the LED built-in directional signals inside the high-altitude package gives you the black leather interior and now if this was an overland it would have or just an overland it would have the overland stitched in there but because you get the high altitude you get the full leather backing there both these seats are heated and cooled driver's side lumbar factory floor mats throughout power windows power locks power mirrors memory driver seat and you get the black rose grain wood grain trim on there the also the black bezels around the handles and the air vents auto headlamps power tilt and telescopic steering wheel you get these seven inch LCD display digital speedometer outside temp and compass really a nice setup and you can adjust pretty much all this stuff in the corner and control it with the steering wheel with these buttons right here leather wrap steering wheel it's heated cruise controls on the right adaptive cruise controls on the bottom there and then Bluetooth and information center controls on the left this one has the 8.4 4C radio I'm going to close the sunshade here on the panel roof so that 8.4 4C radio AM FM serious XM radio capabilities it also has these off-road pages I got it on there just because sometimes it takes a second for those to load but you get your steering angle which is really cool you transfer case if you're in four-wheel drive or two-wheel drive you also have your suspension because it's the quadra lift suspension you can raise it up and you just press it up just press that button right there and you can see it it's actually raising and I don't know if you can see it so now you're in off-road one mode so that's pretty cool stuff there you also have your pitch and your roll your accessory gauges and then your select terrain so now going back to this quadra track 2 system if you do the sand snow mud or rock and you turn that it'll actually change in your seven inch cluster here that sand snow and then mud and then if you wanted to do rock you get to put it in four-wheel drive look we're just gonna go back to auto anyways you also have your factory navigation system on here you can also do Android Auto and Apple CarPlay part of that projection manager right there so that's a really neat system where you can project your cell phone to the screen also music and stuff and then control your cell phone through the screen it's really cool stuff has a backup camera those are dynamic grid lines that turn as you turn the steering wheel and that center line it lines up right with your receiver hitch so if you're hooking up to a boat or a camper or something makes it really easy to do now you got your climate controls here as well you got dual climate controls and you also have your heated and cooled seats and heated steering wheel buttons now if you don't like using the screen up here don't worry you can use the climate controls down here as well as the audio controls for more tactile buttons also we have the perpendicular and parallel park assist that's for your parking sensors you can turn those off and on that's for your lane departure your start stop capabilities your eco mode sport mode and your stability control you do get two USBs and an aux jack underneath there and then you get your keyless entry here with the remote start 8-speed automatic transmission you get that wood grain trim all along the dash there up here you got your home link buttons map lights and this is also your sunroof and sun shade controls so I'm gonna show you that opens up just the front there and then that goes all the way to the back that's a fixed glass rear window we're gonna take a quick look at the back seats kind of running low on time here but back seats lots of room back here you do get the latch child safety system these back seats are heated as well and you get two USBs and 115 volt 150 watt plug-in back here these seats do fold down almost completely flat which is nice if you got a haul a bunch of extra stuff and you do get the child safety locks in the back doors I'm gonna do one more quick walk around here of the car or of the Jeep so you can get a good look and feel for that slate blue color thanks for checking out the video I hope you enjoyed it and learned a little bit about this particular Jeep Grand Cherokee and to see more pictures of this Jeep Grand Cherokee or one of our other 450 new and used cars trucks SUVs minivans Wranglers Grand Cherokees you name it we got it go to our website www.summitauto.com full pictures and descriptions of every single vehicle from two locations all at summitauto.com and if you'd like to check out more HD videos you can go to youtube.com slash summit auto remember to like subscribe and share on this video and all the videos that you see there in fact in a second you will see a link to subscribe to our YouTube channel on your left it links more grand Cherokee videos like this one on your right if you have not been to our website on the bottom link to this vehicle on our website click those check us out and we're super excited to be offering the 2020 Jeep Grand Cherokee overland high altitude edition and slate blue metallic thanks for checking out the video


Apple has asked the U.S. Supreme Court to review a judge's ruling that deemed developers should be able to direct users to alternative payment methods other than Apple's own app store. This follows Wednesday's news of Epic initially also asking the same Supreme Court to weigh in on the ongoing dispute. I personally agree with why Epic is doing this. I'm just kind of wondering at this point what is Epic's play? Are they hoping that public opinion ultimately is going to maybe sway what the court would say? I don't know if that is a great strategy. It's a strategy and we'll just have to see if the Supreme Court even picks this case up.


Halfway through the 2030 agenda, the world is not on track to achieve the sustainable development goals. Climate change, conflicts and economic turbulence have hindered progress in addressing hunger. We need an intense and immediate global effort to rescue the sustainable development goals. Despite multiple challenges, the bond between Rome and New York is stronger than ever. FAO fully committed to support the funding that's solutions by addressing the crucial importance of water to encourage food security. Throughout 2023, FAO has extended its leading role in addressing global food insecurity and making agri-food systems more efficient, inclusive, resilient and sustainable for better protection, better nutrition, better environment and a better life. I think there's never been a time when the world needs FAO in the way it needs us today. FAO's commitment and actions have been clearly demonstrated by its Director General, Chu Dong-Yu's extensive and impactful engagements with heads of state, governments, UN principles and other stakeholders. We need action and work the park. No matter how developed you are, you need to improve your production system. 2023 has also witnessed a sharp increase in core leadership and technical engagements in New York. We have substantially to change the way we use our resources. We must scale up investments in climate resilient communities. With excellence as our standard, our office has continued to deliver our mandate by supporting sustainable development, resilience building, emergency response and broadening stakeholder engagement, including with youth. Member countries should support youth participation in global discussion spaces. Through our dedicated efforts, we have amplified discussions of the water, climate and food nexus, bringing science, technology and innovation hand in hand with indigenous knowledge, emphasizing women and youth voices in the process. Indigenous knowledge needs to go hand in hand with scientific knowledge, that's number one. We need to massively upscale the use of scientific and technological innovations. With a renewed focus on global food security, we will advocate for increased delivery on the four betters, while sharing FAO's new focus on the four R's, recovery, reform, rebuild and renaissance. Together, one UN, one FAO family, one team with one commitment, zero hunger. From all of us at the FAO Liaison Office in New York, we thank you for your continued support and we look forward to working with you in 2024 as we kick off the next phase of our journey together.


This lecture is part of Berkeley Math 115, an introductory undergraduate course on number theory. And this lecture will be about primes. So I'll just recall definition of a prime. First of all, we define it for positive integers. So prime p is an integer greater than 1, not divisible by anything except 1 and p. So of course we know the first few primes of 2, 3, 5, 7 and so on. The letter p is often used for prime numbers. p obviously stands for prime. Notice by the way that we have this condition p has to be greater than 1. 1 is not prime. And the reason it's not prime is that I've defined it not to be prime. And the reason I've defined it not to be prime is that it turns out to be very convenient to have 1 not count as a prime for reasons we will see fairly soon. Not all mathematicians agree with this. There was a mathematician called Laina who used to compute primes a lot and he sort of annoyed all other mathematicians by counting 1 as a prime. So he would produce counts of the number of primes less than a million or a billion or whatever and his counts are always off by 1 with everybody else's count because he thought 1 should be a prime. So that's a prime if you're talking about positive integers. There's an alternative definition where we allow negative integers. So here we want minus 2, minus 3 and minus 5 and so on to count as primes. And here we would say a prime is an integer p not equals 0 such that if p equals a b then a or b is a unit. A unit is something that has an inverse and the only units for the integers are just 1 and minus 1. For more general rings sometimes we will allow other things to be units. In fact we will see a few examples of this bit later. And let's start by recalling simple ways to test for primes. So suppose we want to test some numbers say 101 is prime. We just need to check all the numbers 2, 3 up to 100. These do not divide 101.  First of all we only need to check primes because if it's divisible by say 6 then it would have to be divisible by one of the prime factors of 6. Another way to speed it up is we only need to check numbers less than or equal to the square root of 101. That's because if 101 equals a b and a b are both greater than the square root of 101 then a times b would be greater than 101 which is a contradiction. So we only need to check numbers up to the square root. So we just need to check 2, 3, 5 and 7 because these are the prime factors less than the square root of 101 and we can easily check it's not divisible by any of these. So we can easily see that 101 is prime. You can see the number of steps for this is going to be roughly bounded by about the square root of whatever number we're checking. So if we're checking n to be prime then we're going to need to check about the squares of n numbers. Well a bit less than that because we only need to check primes but whatever this gives a crude bound. If n numbers n up to about a thousand or up to about a billion if you've got a computer this is quite reasonable. This is too slow if n is large and if n is a really large number say it has a hundred digits or something you wouldn't use this test because it's too slow. Well actually you would use this test part of the way because we have some fast ways to check whether numbers are prime so they take a long time to warm up and you know before trying one of these heavyweight tests what you should do is try the easy tests. You check n to see if it's divisible by 2 or 3 or 5 because these are really fast tests to do and if you've got a computer you might for example check that n isn't divisible by any prime octramillion or so before starting one of these more complicated tests. So checking a number for divisibility by a few small primes is nearly always a good thing to do when you first come across a number. Now we have the fundamental theorem of arithmetic which says that any number n, n is greater than equal to 1, n is a product of a finite set of primes in a unique way. Well it's not quite unique because 2 times 3 is equal to 3 times 2 so we say a unique way op to order and you may think you thought of an obvious counter example because you might say that n equals 1 is not a product of primes but I will say sort of note that 1 is a product of the empty set of primes because by convention the product of an empty set of numbers is taken to be 1 in much the same way that the sum of an empty set of numbers is taken to be 0 and it's actually very convenient to have this convention about the product of an empty set of primes being 1 it makes several theorems rather easier to state. So that's a product of primes for positive integers. There's an alternative version if you allow positive and negative integers where you'd say if n is not equal to 0 n is a product of primes. Here we allow primes to be possibly negative in a unique way op to first of all order and secondly op to units. And it's actually not quite correct to say it's a product of primes in a unique way because minus 1 isn't actually a product of primes. So what you have to do is remember to say it's a product of primes and a unit and it's in a unique way op to order in units so you can have 2 times 3 is equal to 3 times 2 is equal to minus 2 times minus 3 and so on. So these two versions are pretty obviously equivalent. I'm just going to prove this version here and the second version follows immediately. So the proof is in two parts. There's an easy part which is existence of a factorization n equals p, q, r and so on where these are primes and there's a hard part which is uniqueness which says that this is unique op to order. So let's do the easy part. Let's take a number n. If n equals 1 or n prime then we're done because it's a product of an empty set of primes or a product of just 1 prime. If not, otherwise n is equal to a, b where a and b are both greater than 1 and less than n because if you can't write n like this then it must either be 1 or a prime more or less by definition of a prime. So by induction a is a product of primes p1, p2 and so on and b is a product of primes q1, q2 and so on. That's because these numbers are less than n so we can apply our inductive hypothesis about every number being a product of primes and so obviously n is equal to p1, p2 and so on times q1, q2 and so on. So by induction on the integers we've shown that every integer is, every positive integer is a product of primes. Now we've got to do the hard part, uniqueness and here we need a key property of primes so this is the key step.  So this is the central key result we need in order to prove uniqueness so how do we prove this? Well suppose p does not divide a then the greatest common divisor of a and p is equal to 1 as p is prime. Now p is prime so the only possible divisors of p are 1 in itself and we've just assumed it doesn't divide a so the only number that can divide a and p is 1. So by Euclid ax plus py is equal to 1 for some integers x and y and now we multiply by b so bax plus bpy is equal to b. Well p divides this and it divides a b because we assumed it divides a b and p obviously divides this because p divides p so p divides b which is what we're trying to prove we've shown that if p doesn't divide a then it must divide b. So with this key step it's now easy to prove uniqueness. So let's suppose p so let's suppose n is equal to p1 p2 and so on product of primes and suppose this is also equal to q1 q2 q3 and so on. Then p1 divides q1 q2 q3 and so on. And now we show that if p1 divides a product of two numbers then it must divide one of them and obviously you can just sort of repeat this by induction and show that if a prime divides any product of numbers then it must equal then it must divide one of them. So p1 divides qi for some i but then qi is prime so p1 equals qi. And now what we can do is we can just go back and p1 is some qi here and we can just cross off p1 and qi from the factorization and repeat. So we sort of repeat this after dividing by p1 equals qi. So the factorization is unique up to order. Of course we can't insist on the order because p1 doesn't necessarily divide q1 it just has to divide one of these. So this proof of uniqueness of factorization into primes is more or less in Euclid's elements of mathematics written more than 2,000 years ago. He didn't quite state it and the reason is that the Greek mathematical notation was so clumsy that it was actually really difficult to even state the product. State that n can be written as a product of primes. The trouble is the Greek notion of multiplication was a bit tricky. They tend to represent numbers as line segments and you can multiply two line segments to get an area and three line segments to get a volume. But if you haven't yet discovered four dimensions it's a bit difficult to do a product of four numbers geometrically. So what did Euclid do? Well what Euclid did was he proved the key step here. He proved that if p is a prime and divides a b then it divides a and it divides b and that's the hard part of proving the fundamental theorem of algebra. So Euclid is sort of credited with proving this theorem because he gave the essential hard part of the proof. We can also give a similar result for polynomials over say the reals or any other field. So if you've got a polynomial f of x is a0 plus a1 x and so on. Then we can talk about polynomials being prime if you can't write them as a product of polynomials of smaller degree. Notice here the units are the constant polynomials that are not identically zero. So the result here says that any polynomial, any non-zero polynomial can be written as a product of irreducible polynomials and constant polynomials. And this is unique up to order and multiplying everything by constants. I'm not going to give the proof of this. You more or less just copy the proof I gave for the integers. Now I want to give some examples to show why we actually need to prove this and why it's not entirely obvious. So for example you can ask a similar result in any collection of objects that you can multiply. We can multiply integers so we can define primes and talk about unique factorization. If you've got any other collection of objects you can multiply you can again try and define primes as things that can't be factorized and so on. So let's look at some cases. Suppose you look at the reals that are greater than or equal to one. Again you can multiply two reals that are greater than or equal to one and you will get another real. There's one unit which is one. And what would a prime be? Well a prime would be something that can't be written as a product of two things that aren't units. But the problem is if you take any real you can write it as the product of the square root of A times itself. That's any real greater than one. So there are no analogs of primes. Let's put primes in inverted commas. So here you can't write a real number greater than one as a product of a unit and primes. Where by unit and primes we adjust the definition for the reals greater than or equal to one. A similar example, well the reals are sort of continuous in some sense. And we can ask can we do something like this for things that are discrete? Well here now let's take the integers 1, 5, 9, 13, 17, 21. So these are the integers of the form 4n plus 1. And notice you can multiply them because 4n plus 1 times 4m plus 1 is equal to 4 times mn plus 1. So here we can multiply these integers and you can ask what are the primes here? Well this one is number 1 is obviously just a unit and 5 is prime and 9 is prime. I mean you can multiply 9 as 3 by 3 but that doesn't count because you're only lying numbers that are one mod 4. So these are all primes, 21 is again prime but 25 is not prime and 29 is prime again and so on. So 25 is not prime because it's 5 times 5 and similarly 45 would count as not prime. So we have an analog of primes and we can write every number is indeed a product of primes. And I'm going to write primes in inverted commas because I'm talking about these funny primes where I'm counting 21 and 9 as being prime. However this is not unique. And the problem is we can do things like 9 times 49 is equal to 21 times 21 and 9, 49, 21 and 21 are all primes in this funny sense.   So we restore unique factorization if you add the numbers that are 3 mod 4. So anyway the point of this example shows that you actually have to do some sort of non-trivial proof that ordinary integers can be written as a product of prime. So you've got to eliminate some sort of weird case like this going on. Well the problem with these two examples you could say the reals greater than or equal to 1 and these numbers are not actually closed under addition and subtraction. So can we find examples with addition and subtraction where we don't get uniqueness? And the answer is yes. So let's look at say functions on the reals that are greater than or equal to 0. And here we can multiply them and add them and subtract them point wise. But if you look at the function x you can see x is equal to root of x squared and root of x is equal to the fourth root of x. All squared and the fourth root of x is equal to the eighth root of x all squared and so on. So you can keep on splitting this function into a product of smaller and smaller things and none of these are units. So there's no way to write this as a sort of product of prime functions because we can always split the function as a product of two smaller things. So we can't write functions as a product of... If you're looking at general functions on the reals there's no good concept of a prime function that you can write all functions as a product of. Another example, let's take all numbers of the form m plus n times the square root of minus 5 when m and n are integers. This is an example of something called an algebraic number field and a lot of number theory can be extended from the integers to algebraic number fields. However, one thing that can't be extended in this case is unique factorization because we can see that 6 is 2 times 3 and it's also 1 plus root minus 5 times 1 minus root minus 5. And this gives two different factorizations of 6 and they don't differ from each other by units and you can't write either of these as a product of none units. So there's no uniqueness of factorization in this case. Incidentally, we will see later on that n plus n, I think the numbers of the form n plus root minus 1 do have unique factorization into primes. These are called the Gaussian integers and we will later adapt Euclid's proof to show that it can be extended to these as well. Here the units by the way are plus or minus 1 and plus or minus i, which is root minus 1. So it has slightly more units than the integers. So in some sense, we're quite lucky that the integers do have unique factorization. I mean, we could easily have ended up with the integers behaving like these numbers here, which would make life a lot more difficult. Next, we will have Euclid's proof that there are infinitely many primes. I discussed this in the first lecture, so I'll just quickly recall it. So you remember what we do is if we've got any finite set of primes p1 up to pn, what we do is we just multiply them together and add one. And what we do is we take a new prime p that divides this, and then we notice that p is not equal to p1, p2, up to pn, because if it was p1, say, then it would divide this and it would also divide this, so it would have to divide one, which is nonsense. So p is a new prime. So there are infinitely many primes. Well, Euclid didn't actually say there are infinitely many primes because he didn't like the concept of infinity. So what he said is that for any finite set of primes, you can find another prime not in that set, which is another way of saying that. Of course, he didn't use the word set because set theory hadn't been invented, but whatever. As I said in the first lecture, there's a common mistake that some people think that p1 up to pn plus one is always prime, and you can easily check that it isn't. I mean, let's do the first few cases. First of all, we take the product of the empty set of primes, which is one and add one, and we get two. Then we take the product of all the primes we found, two, and add one, which is three. Then we take two times three plus one, which is seven. And we take two times three times seven times plus one, which is 43. Then we take two times three times seven times 43 plus one. And this time we get 1807, and this factors as 13 times 139. So we don't always get primes. We really do sometimes get this number factorizes. There's something you might wonder about. What happens if instead of taking two times three times seven times 43, you just take all the first few primes, two times three times five times seven times 11, or whatever, and add one? Is this always a prime? See, it looks as if it's got a much bigger chance than normal of being prime because it's obviously not divisible by all these small primes. So these numbers are certainly more likely than most numbers to be primes in a sense. However, even these are not prime. We should check the first few cases and just see what happens. So we take two times one. That's obviously prime. Two times three plus one. That's seven. That's prime two times three times five plus one. That's 31. Two times three times five times seven plus one. Well, that gives you 211. That's still prime. However, if you get two times three times five times seven times 11 plus one, that's again 2311. That's again prime. If you take two times three times five times seven times 11 times 13 plus one, this is 30031. And this actually splits as 59 times 509. So the reason I'm doing this is to sort of emphasize that number theory is really a rather experimental subject. When if you've got a question like is a product of the first few primes plus one is prime, the first thing you do is just calculate the first dozen or so examples and see what happens. And quite often you will be able to answer your question by just doing a bit of calculation. We can have another question. Suppose you take Euclid's method of generating prime. So you remember we get two, three, then we got seven, then we got 43, then we got 13 and so on. And you can ask, do we get all primes appear in this list? And this is actually an incredibly difficult question. Well, in some sense it's a very easy question because it's easy to see that the answer is almost certainly yes. The answer is almost certainly. And the reason for this is as follows. Suppose we take some prime, say, does 101 appear? And let me try and convince you that it is almost certain that 101 appears without actually doing any calculation. So the chance that it appears at step n is about 1 over 101 because we're dividing a sort of random number by 101 and the chance that a random number is divisible by 101 is about 1 and 101 because there are 101 possible one sorts of remainders. So the chance that it does not appear at step n is about 1 minus 1 over 101. So the chance that it does not appear at step less than or equal to n is about 1 minus 101 to the power of n. And you can make this as small as you like by making n sufficiently large. So if you take n to be a million, this will be a ridiculously small number. So it's almost certain that 101 appears in the first million numbers. So this is a bit like saying, you know, suppose you toss a coin a million times. What's the, you know, it's very unlikely it's going to turn up heads every time.  But if you threw it millions of times, it's as certain as almost anything can be that this number is going to turn up somewhere. So the number 101 almost certainly appears. It's very hard to imagine how it couldn't appear. And the same obviously applies to any other number. So this strongly suggests that all primes do in fact appear. However, it seems to be almost impossible to prove. You see, we're saying these numbers are random. Well, the trouble is they're not random. They're produced by this, you know, Euclid's method and, you know, there's a specific computer program. And it's conceivable that there's some very subtle structure that we don't know about that somehow forces all these numbers to be not visible by 101. And I've no idea how you could possibly prove this. So this is an example of a statement in number theory that it's almost obvious that it's true, but we have just no idea how we can actually prove it. I think someone once said number theory is a subject where any fool can ask a question that nobody knows how to answer. And this is a typical example of such a question. Number theory is full of questions like this. Now we have another question. We've shown there are infinitely many primes. And if you look at the primes, you notice that the last digit of a prime must be 1, 3, 7 or 9, unless the prime is 2 or 5. Because if it's not 1, 3, 7 or 9, the number would have to be divisible by 2 or 5. And if you write that out a lot of primes, you see the last digit seemed to be roughly equally common. So we can ask, are there infinitely many with last digit 1? In other words, are there infinitely many primes with a form 10n plus 1? And the answer is yes. This is a very famous theorem proved by Dirichlet, which we will actually discuss later in the course. What I'm going to do now is to give a few easy cases of this, or easy variations. So instead of 10n plus 1, we could obviously ask for any other arithmetic progression. We could ask, suppose we're given numbers a and b, are there infinitely many primes with a form a and plus b?  And then I guess we should also say a is not equal to 0, otherwise you could just take 0 times n plus 1 or something like that. So there's some obvious easy cases. If you look at primes of the form 2n plus 1, there are obviously an infinite number because all primes except 2 are of this form. So what about numbers of the form 4n plus 3? Well, what you can do is you can do a sort of variation of Euclid's proof. What we do is suppose we take some primes of the form p1, p2, up to pk of the form 4n plus 3. Then I'm going to multiply them all together and then I'm going to multiply this by 4 and subtract 1. So this is of the form 4m plus 3, obviously. And we notice this is not divisible by primes p1 up to pk that we found because if it were then these primes would divide this number and they would divide that number and so they would have to divide minus 1. However, it's also not divisible entirely by, let me rephrase that. It must have at least one factor not of the form 4m plus 3, sorry, not of the form 4m plus 1. That's because the product of any number of factors of these is also of the form 4m plus 1 as we saw earlier. If you multiply numbers of the form 4m plus 1 it's still of the form 4m plus 1. So this must have at least one factor that's not of the form 4m plus 3 and it can't be divisible by 2 either. So it must have a prime factor of the form 4m plus 3. And it can't be one of the primes that we've already found of the form 4n plus 3. So there are infinitely many primes of the form 4n plus 3. We can do a similar thing for 3n plus 2. And I'll just leave this as an exercise. There are infinitely many primes, greater than positive primes of the form 3n plus 2. What about 4n plus 1? Well this is rather more difficult. The problem is that the product of 2 primes of the form 4n, numbers of the form 4n plus 1 is of the form 4n plus 1 which we used earlier. The trouble is if we multiply numbers of the form 4n plus 3 together they don't need to be of the form 4n plus 3 so the previous proof breaks down. Well we can do 4n plus 1 but we need to use the following fact. That if a prime divides m squared plus 1 then p equals 2 or p is of the form 4n plus 1. And what we're going to do is we're going to prove this later on in the course but for the moment I'm just going to assume it. And as I said whenever we have something in number theory you should always start out by checking the first few cases just to get an idea what's going on. So let's look at the numbers m squared plus 1.  That's of the form 4n plus 1. 3 squared plus 1 is 10 which is 2 times 5. 4 squared plus 1 is 17. That's of the form 4n plus 1. 5 squared plus 1 is 26 which is 2 times 13. That's of the form 4n plus 1. 6 squared plus 1 is 37. That's okay. 7 squared plus 1 is 50. So that's 2 times 5 times 5. And you see all the numbers that appear here are always either 2 or of the form 4n plus 1. So I've at least made this statement reasonably plausible. Anyway, now using this fact we can now copy Euclid's proof. Suppose we take some numbers p1, p2 and so on up to pk where all pk's, we're all pi's of the form 4n plus 1. What I'm going to do is I'm going to multiply them together and I'm going to add 1. But that's no good because I can only show that something is of the form 4n plus 1 and it divides a square. So I can square this. But then the problem is this might be divisible by 2. So I'm going to get rid of the possibility that this is divisible by 2 by sticking in a 2 here. So we now know that all factors of this are First of all, they're of the form 4n plus 1 by the result that I postponed the proof until later that I stated earlier. Secondly, they're not equal to 2. Thirdly, they're of the form 4n plus 1 or 2. And I've shown they're not equal to 2 because I stuck a 2 in here. And they're also not equal to p1, p2, up to pk. So they must be new primes of the form 4n plus 1. So we've got infinitely many primes that are 4n plus 1 and infinitely many that are 4n plus 3. Well, you can't really push this rather simple idea much further. You can push it slightly further so we can show that infinitely many primes that are 3 or 7 that have lost digit as 3 or 7 when written to base 10. And the point is that if m and n have lost digit 1 or 9, so then so does the product. That's m can have lost digit 1 or 9 and n can have lost digit 1 or 9. And if you multiply them together, the result will still have lost digit 1 or 9. So if you take something like 10 p1 up to pk plus 3, it must have a prime factor with lost digit 3 or 7. So if you take p1 up to pk to be all the primes you've thought of so far lost digit 3 or 7, then a prime factor of this will be a new one. The trouble with this is it's very hard to separate these out. So if you want an infinite number with lost digit 3 or an infinite number with lost digit 7, it's much harder to prove. But this is what Dirichlet managed that we will discuss later. Another question you can ask is what about gaps between primes? So if you look at our primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 27, 29, we can see that we sometimes get gaps. So here's a gap of 6, here's a gap of 4, here's a gap of 1, here's a gap of 2 and so on. So the gaps seem to be, I mean, they're sometimes quite small, they sometimes go down to 2 and they sometimes get a bit big. And we can ask, is there a bound to the gaps? And the answer is no, there's no bound on the gaps and this is quite easy to see. Let's just take the number n factorial and we can add 2 or 3 or 4 or up to n. And these are not prime. And that's obvious because n factorial plus k is divisible by the number k if 1 less than or equal to k is less than or equal to n. So here we've got a gap of n minus 1 numbers that are not prime. So we notice here n minus 1 is slightly less than the log of n factorial. And in fact it turns out that in some sense the average size of a gap between numbers, so if you've got prime numbers up to n, the average size of the gap between prime numbers is about the logarithm of n. That actually follows from something called the prime number theorem that I'll discuss later in the course. And it's incredibly difficult to show that the gap is sometimes significantly bigger than this or sometimes significantly smaller than this. So I'll give a couple of examples. There's an amazing breakthrough by Zhang that the gap is sometimes less than or equal to 70 million for all large n. Well of course it's obvious the gap is sometimes less than 70 million because we've found lots of gaps less than 70 million. What Zhang showed is that you get infinitely many gaps that are less than 70 million. In other words if you keep going forever you will never get the minimum size of the gaps between primes don't keep increasing. It's believed that there are infinitely many cases when the gap actually is size 2 that's the famous twin prime conjecture but that seems to be way beyond anything we can prove at the moment. This number here has since been reduced to something a bit more reasonable than 2 or 300 or something and will probably come down a bit more. On the other hand you can ask can the gap sometimes be bigger than log of n. There's a sort of conjecture that the gap is sometimes about log of n squared so that's about the square of the average size. This seems to be far beyond anything we can prove. Let me give you an example of some of the things we can prove. There's a result by Rankin who found it in about 1938 which says the gap is sometimes bigger than this rather spectacular. You take a third of log of n so that's the sort of log of n is the average size and then you multiply it by log of n then you multiply it by log log log log of n and then you divide all this by log of n sorry log log log of n all cubed and your reaction when you look at this is probably to say this number is utterly and completely ridiculous and you're right. Actually the thing is log of n increases very slowly. Log log of n increases so ridiculously slowly that it's constant for all practical purposes and these two terms are even more ridiculous. However, analytic number theory is a really fond of long chains of logs like this. Chains of three logs occur reasonably often. Chains of four logs like this are unusual even for analytic number theorists so this probably comes close to setting the record for bizarreness of bounds in analytic number theory. This has actually been improved very slightly but improvements on this are actually extremely difficult to do. So I think I'll pause there and continue the rest of the lecture on primes in the next video.


I recently went to a protest in London. So good afternoon, today we are at the March for Medical Freedom, which was a five mile march through, not central London but through London, amazing, amazing how many people are here and just a few words on the intent and I think the title of this, so it's a march for medical freedom and what people are generally saying is no compulsory vaccinations, which actually is the primary, I think, thing that you can see the signs around here, it's mostly about vaccinations. So, medical freedom, if you don't have medical freedom you don't have freedom and if you can't decide what is injected into your body, you are no longer autonomous, you have no autonomy at all and the people here realise this, this is this, this vaccination is to me and I think to many people here an indicator of the level of government control now. There is a real sense here that people know they are losing their freedom and they know that this is a method by which they are losing their freedom. If the government can exercise such control over you as to inject you with a serum and an unlicensed, pretty much untested, uninsured vaccine, then look at the level of control they are exercising over us. They are exercising this control via fear, which is almost always how governments exercise control fear. They are frightening people. I'm walking around London today, I've got to say, heavily, heavily, a high percentage of people still wearing masks, higher to me than other parts of the country that I've seen. I think there's a couple of reasons for that. One is, on this way it sounds cheap, but actually it's true. London is very left wing nowadays and the level of compliance among left wingers is actually higher these days than it ever has been, but the fear is large in London as well. You can feel it and you can see it on the tubes, you've got posters around telling you to wear your face mask. I've seen it on the sides of buses, I've seen it on passing buildings, where your face mask. We've just come through Brixton and we've passed an Iceland where I used to shop actually when I lived around here and in the window where it signs on every window asking people to wear face masks and this is no longer compulsory but people are still being asked to wear it and the level of compliance seems sadly quite high in London, something I've noticed walking around but the people here get this, they really do get this, they know their freedom is being taken away they know that it isn't just about a vaccine, it is about this level of control, an extraordinary level of control. A few words to and about the mainstream media, I have this sneaky suspicion that today if reported at all will be reported as a few hundred people, this is not a few hundred people, this is thousands of people and the mainstream media does this to try and persuade us, to try and persuade you people out there that this is a marginal movement, it isn't, it's big but it must be bigger and it should be bigger and the irony of it all is that it would be better, bigger if not for the mainstream media that there are people, I know there are millions of people sitting at home watching the mainstream media thinking that this is a marginal crowd but believing that maybe they've got a point people who are, the only word for it is lonely, people who are lonely, who think that they are alone in all this I really would encourage people to come to these marches and to come to these rallies because they are bigger than you think, they are hugely diverse, I don't like to use that word but in a good way, there are people from all different walks of life here and they all know that something very sinister and dark is going on and that sinister and dark is that they are losing their freedom there are millions of people at home who know this as well, they might not even be able to articulate it to themselves but they are not on their own and if you are one of those people I can assure you you are not on your own come along, you will see, these are not tin foil hat conspiracy theories these are people from all walks of life who know that their government is taking away their freedoms if you feel like that at all, if there's even a part of your mind that thinks something not quite right here come along, find out more and find out that you are not on your own and find out that this is real, they are really taking our freedoms away confront them, join with other people and stand up, stand up to this you are not on your own, just look behind me and you will see that you are not on your own this is part of the great reset, the great reset is a change, dramatic change in the relationship between the people and the government it isn't confined to coronavirus though, climate change is part of the great reset and yes, mass migration is part of the great reset now if only we could get this many people out protesting against that level of control that change, that great reset then we'd be in a much better place but it is really encouraging to see this many people out here it's been a great energy, it's been a great day and it's not over yet people were marching against forced vaccination and so-called vaccine passports it was attended by thousands of people from all walks of life there was an incredible atmosphere and people certainly seemed to be having a good time and who can blame them, we're just coming out of a long period of house arrest on top of this, the mainstream media and the politicians will only allow one narrative on vaccination to be heard and that is that the vaccines are wholly positive and will give us back our freedom this is absolute nonsense of course, and millions of people know it but they are not represented in the media and many will no doubt feel completely alone in thinking what they think protests allow people to get together and talk about these issues freely it's very important that we can do that but I couldn't help but wonder at the time just imagine if all these people stood for election protests are great and they're important but the hard truth is that politicians don't pay attention to them the hard truth is that politicians will only pay attention when their cushy, highly paid, high status jobs are threatened that's when and only when they'll sit up and take notice the only true power is at the ballot box and the only thing politicians care about is the ballot box let's take the EU as an example for years people had wanted to leave the EU many people had campaigned for it for a long time but it was only when UKIP began winning seats on councils and in the European Parliament that the mainstream politicians began to care if UKIP hadn't threatened the Tories at the ballot box there would have been no EU referendum think about that for a second you'll realise that it's true I greatly admire those who go out and protest I really do but if those people really want to achieve their aims they must put themselves forward stand for election and win elections that's the only way I am therefore asking those protesters to join for Britain if you want your freedom back you must realise what threatens it Covid yes, but not solely Covid the threats to your freedom come from a variety of sources and you must fight them all continue to protest by all means but power sits in the elected chambers and it is therefore vital that we sit in the elected chambers join for Britain, stand for election take back your freedoms don't rely on the politicians who took them away to give them back to you it doesn't work that way this is the only true option make your voice heard loud and clear and allow the British public the chance to vote for the restoration of their liberties and yours join us at www.forbritain.uk thank you thank you for watching if you would like to stay updated with all of our latest videos please like the video below and subscribe to our channel as you probably know Forbritain is shadow banned on most social media so it's really important to like and share our videos in order to get our message out thanks again for helping us to fight back thanks again for helping us to bring back Britain