2024/05/10 by Craig, William, van Ittersum, Jan-Willem, Ono, Ken · 4 citations
#05A17 #11Fxx #11P81 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2405.06451
We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer n≥ 2 is prime if and only if (3n3 - 13n2 + 18n - 8)M1(n) + (12n2 - 120n + 212)M2(n) -960M3(n) = 0, where the Ma(n) are MacMahon's well-studied partition functions. More generally, for "MacMahonesque" partition functions M_a(n), we prove that there are infinitely many such prime detecting equations with constant coefficients, such as 80M(1,1,1)(n)-12M(2,0,1)(n)+12M(2,1,0)(n)+…-12M(1,3)(n)-39M(3,1)(n)=0.