arXiv:2101.08038v1 [math.NT] 20 Jan 2021 ahmtclAsc fAeiaAeia ahmtclMont Mathematical American America of Assoc. Mathematical htteeaeifiieymn ar fpie fteform the of primes in of For pairs conjecture. many prime infinitely twin are the there is that fields finite over olution over a rcsl on h number the count precisely can oil ntelte etn samncirdcbepolynom more irreducible significantly is monic polynomials a prime of is distribution setting the latter the factorizatio in unique nomial) being both including features, common uniaiefr fti ojcuedet ad n Little and Hardy to due conjecture integers distinct this of form quantitative pn ept h pcaua ratruh fZag[ Zhang of breakthroughs spectacular the despite open, aur 2014] January structions. o hc h imn yohssi qiaett h assert the c to equivalent of is problem hypothesis classical Riemann the the with which contrasted for be should this and n .INTRODUCTION. 1. where oe.Tern fintegers of The power. as as fpienmesls reulthan equal or less numbers prime of + nntl ayTi rm oyoil of Polynomials Prime Twin Many Infinitely x x ifrc rmsadt rtmtcpoete ftecombin the rel of its properties and arithmetic degree, to odd and of primes di tuples Wieferich we prime spirit, twin same of the family In infinite Pollack. and Hall by cases different ifrb xdcntn,freach for constant, fixed a by differ Abstract. ffiiefils hr r nntl aypiso oi irred monic of pairs many infinitely are There fields: finite of nte aos ogsadn pnpolmo ubrtheor number of problem open long-standing famous, Another a ∞ → F ∞ → 1 q µ n , . . . , swsdn yGus h rvdthat proved who Gauss, by done was as , ( d ) . o ongtv constant nonnegative a for , hl h wnpiecnetr ssilfmul pn tho it open, famously still is conjecture prime twin the While stecasclMoisfnto.A eut as result, a As M¨obius function. classical the is + a a 1 a , . . . , r π is prime simultaneously are lieBri n ate Issac Matthew and Burrin Claire ( x ; Let a r π h number the , 1 ( a , . . . , x π F Z WNPIEPOLYNOMIALS PRIME TWIN = ) q π q ( d Degree Odd q n h oyoilring polynomial the and eafiiefil of field finite a be n π ( = ) n q q log ( = ) r l 2: aur 1 0114 ..mnhyv.e ae1 page monthly-v4.tex a.m. 1:49 2021 21, January 121:1 hly n ≥ x ) x ) ∼ is q x 3 n S fmncirdcbeplnmaso degree of polynomials irreducible monic of n lmnay osrciepof eegvnfor given were proofs constructive Elementary, . π n 1 + S ( ( + a X x d ( O 1 | a ; n O a , . . . , a 1  µ a , . . . , 1  x a , . . . , ( d 1 q / q ) n/ n 2+ q r ≥ 2 n ) d o r  cs h osrcino further a of construction the scuss tra elnumbers. Bell atorial (1) 15 , noiglclcnrec ob- congruence local encoding ) tost h xsec fcertain of existence the to ations r 3 cbeplnmasover polynomials ucible ) (log ,  n anr [ Maynard and ] lmns where elements, fintegers of F opee osatwt,one with, start To complete. , oan.Apie(poly- prime A domains. n ( q x q a.Orudrtnigof understanding Our ial. ,p p, [ o httenumber the that ion x X n utn rm numbers, prime ounting ees h ojcueis conjecture the tegers, odasrsta given that asserts wood ) ∞ → ] r d rei h setting the in true lds 2) + xii ubrof number a exhibit ihahpirres- happier a with y n , n hsi still is this and ≤ 7 q x .Arefined A ]. saprime a is o which for F q that π ( x n 1 ) Mathematical Assoc. of America American Mathematical Monthly 121:1 January 21, 2021 1:49 a.m. monthly-v4.tex page 2

To formulate the corresponding problems over finite fields, let q ≥ 3. The size of a nonzero polynomial f of degree n over Fq is defined to be

n |f|q := q = |Fq[X]/(f)|.

Two prime polynomials f, g ∈ Fq[X] form a twin prime pair if the size of their dif- ference |f − g|q is as small as possible, namely, if |f − g|q = 1. The twin prime conjecture asks for the existence of infinitely many twin prime pairs (f,f + a) for × some fixed a ∈ Fq . Given positive integers n and r, and given distinct polynomials a1, . . . , ar ∈ Fq[X], each of degree less than n, the Hardy–Littlewood con