prime number theory prime prime number divides connecting little states odd prime divides described works pertaining time known connections topics mathematics discovered including types numbers numbers specific types types numbers original definition prime time connections discovered extended cover properties certain prime numbers general subjects number fields abc october known sequence equivalent definitions stronger version little prime usually expressed relation definition relation follows property equivalent definition given beginning prime prime divides formula following examples formula leaves remainder division prime formula intermediate omitted clarity leaves remainder division prime defined equivalent prime sides raising sides power shows prime true implies order divides prime implies defined orders coincide way proved formula history search status proved solutions later decade showed specifically case solutions odd prime prime satisfy words exist solutions odd prime examined formula asked question tried find expressions answering question prime found prime confirmed prime calculated smallest formula found zero providing later ordered elementary calculations inspired earlier work simplified proof showing remarked factor shown possible prove prime complex numbers contrary method hinted aware proof complex values prime found prime proof prime published guy doubled previous record set extended search aid able reach search limit limit extended finally reaching known exist greater search new currently performed distributed computing project december search started project november extended search limit continues wilson infinitely exist number approximately log log result follows plausible assumption prime degree roots unity uniformly distributed group properties connection following connecting proven let prime let furthermore assume divide product prime case divide commonly known case said fail prime solutions equation exist holds expanded showing hold true prime divide proved actually divide prime extended proof let set pairs greatest common prime power ideal defined sin field extension obtained adjoining number field rational numbers extension known number field particular case root unity number field ideals follows case solutions divides elements showed prime connection abc prime prime satisfying showed abc holds exist infinitely precisely showed abc implies existence constant depending number base equal variable greater log goes infinity numerical evidence suggests prime numbers given interval set set denoted respectively sets shown finite necessarily infinite proper set prime numbers later shown existence infinitely follows weaker version abc called additionally existence infinitely follow exist infinitely square free numbers exists real number set density index composition defined formula formula meaning formula gives product prime factors connection known number prime prime little implies prime numbers prime prime prime prime prime divides prime prime notable open problem determine numbers prime index square free prime number square free exists prime divides prime numbers prime index square free showed related result infinitely square free numbers infinitely similarly prime divides number prime fact exists natural number prime divides formula formula prime example divides formula divides formula numbers special situations formula formula square free formula formula fact exists prime divides formula prime clearly formula prime prime notice odd prime divides formula divides period length divides formula prime period length times shown number prime index number prime powers connection scott showed equation solution positive denotes order showed solution equation specific set hold prime greater johnson observed known greater numbers periodic expansions home project searches testing numbers greater number periodic expansion bit pseudo length tested numbers generated combination bit strings bit length found new prime noted sequence known greater mutually friendly numbers shared index connection observed known square factors non square free later showed repeated factors addition following connection exists let base prime formula formula furthermore prime connection directed cases doubling gives number cycle term doubling refers directed natural numbers arrows pointing reduced shown odd prime numbers properties related number fields shown formula formula odd prime formula fundamental imaginary field formula furthermore following shown let prime let formula fundamental imaginary field formula let formula fundamental imaginary field formula formula formula context denote furthermore following result obtained let odd prime number order formula assume divides class number real field formula field obtained adjoining sum root unity field rational numbers prime holds conditions replaced condition replaced case wall sun sun prime condition replaced prime satisfying small commonly called near prime sequence near represent recent searches addition primary search tried find near following table lists near interval search bound reached search effort sign easily predicted criterion second supplement law different definition near prime defining prime small value formula formula respect operation gives smallest absolute value following table lists formula notions related follows formula clearly formula chosen formula small clearly formula small number formula odd related small example formula formula reads extremely non near formula near second definition base prime base prime prime divide divide natural number infinitely base showed positive setting leads shown known solutions small values checked information note solutions union prime divide solutions smallest solutions term sequence known solutions infinity solutions natural number bases prime solutions shifted solutions set solutions sequence base prime prime sequence consider formula formula little formula true prime natural number natural number infinitely formula known solutions small checked pairs pair pair satisfy prime form pair known instance case known pairs sequences sequence start natural number smallest prime divide divides solution trivial solution natural number makes sequence periodic example let gets cycle let gets cycle let longer sequence gets values unknown status example let known base let known prime base divides let longer sequence gets unknown values exist resulting sequence eventually periodic start value unknown numbers number odd natural number satisfying denotes function according odd natural number number prime prime numbers sequence shown numbers particular exist exactly numbers matches number numbers currently known generally natural number number base definition number odd natural number formula order numbers sequence number prime prime lucas lucas prime associated pair prime denotes lucas sequence kind equals symbol formula associated pair let natural number called wall sun sun called called example prime prime wall sun sun prime fact prime like traditional base prime places let global field number field function field variable finite field let curve non place norm called place base place base strong place base order gives number rational points field reduction
Wieferich prime