A Monic Sextic Polynomial Generating 36 Consecutive Prime Values

We present the monic sextic polynomial

P(n) = n^6 − 83n^5 + 2857n^4 − 52217n^3 + 534664n^2 − 2909420n + 6577037,

which generates 36 consecutive prime values for n = 0,1,…,35 in canonical form (positive primes only, starting at n = 0).

The sequence stops at n = 36, where

P(36) = 115278749 = 701 × 164449

is composite.

Moreover,

P(−1) = 10076279 = 167 × 60337

is composite, so the polynomial has no effective child under the translation n → n−1 and is therefore a genealogical peak.

This example is particularly relevant in the framework of polynomial genealogies, as it provides a high-degree (sextic) canonical peak with length L = 36 that does not exhibit palindromic symmetry in its prime sequence.

The full list of generated primes and the complete translation genealogy from L = 1 to L = 36 are included.