A Monic Sextic Polynomial Generating 36 Consecutive Prime Values

We present the monic sextic polynomial P (n) = n⁶ − 83n⁵ + 2857n⁴ − 52217n³ + 534664n² − 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.