A Monic Cubic Polynomial Generating 40 Consecutive Prime Values, Counted by Absolute Value

Abstract

We prove that the monic cubic polynomial\(P(n) = n^3 + 54n^2 - 2917n + 20021\)has the following properties: \(|P(n)|\) is prime for every integer \(n\) with\(0 \le n \le 39\), and \(P(40) = 53741\) is composite. The forty values containexactly thirty-nine distinct primes, since \(|P(17)| = |P(19)| = 9049\). Thepolynomial is a genealogical peak, since \(P(-1) = 22991\) is composite. Wedescribe the search method that produced \(P\). It is based on a small-valuecondition on a fixed finite set of indices together with the genealogicaldynamics of monic polynomials. We also give a verification in PARI/GP. No claimof absolute optimality is made.