Q41 Polynomials (L = 41)

The polynomials

f(n) = n⁴ − 80n³ + 1817n² − 8680n + 87119

g(n) = n⁴ − 80n³ + 1911n² − 12440n + 188369

These monic quartic integer polynomials produce a run of 41 consecutive prime values when evaluated at consecutive integers starting at n = 0. As usual in this context, primality is tested on the absolute value P(n).

Run length: L = 41

Category: monic quartic prime-generating polynomial

Both polynomials are monic quartics, since the leading coefficient (the coefficient of n⁴) is equal to 1. This places them within the class of monic quartic prime-generating polynomials, a mathematically significant and more restrictive family than generic quartic polynomials.

The achievement of L = 41 exceeds by one the classical run length 40 obtained by Euler’s quadratic polynomial, while being attained here within the stricter class of monic quartic polynomials.

Independent verification (online)

Both polynomials were independently verified online by scanning consecutive integers from n = 0 and stopping at the first composite value. The SageCell outputs were:

• STOP n = 41, P(n) = 97697 for f(n)
• STOP n = 41, P(n) = 202801 for g(n)

First composite values at n = 41

• f(41) = 97697 = 151 × 647
• g(41) = 202801 = 139 × 1459

Snippet SageMath (SageCell) (as used in online verification)

def check_run(P, N=120):
    for n in range(0, N+1):
        x = abs(P(n))
        if not is_prime(x):
            print("STOP n=", n, " P(n)=", x)
            return
    print("No stop up to N =", N)

# Q41-A
check_run(lambda n: n^4 - 80*n^3 + 1817*n^2 - 8680*n + 87119)

# Q41-B
check_run(lambda n: n^4 - 80*n^3 + 1911*n^2 - 12440*n + 188369)

Genealogical origin (translation n → n − 1)

Within the Borghi genealogy framework, a translation step is defined by: Q(n) = P(n − 1). If P(−1) is also prime, the prime run for Q(n) starting at n = 0 increases by one.

Connection to Strategy A seeds (L = 39)

Both L = 41 polynomials arise as descendants of quartic seeds found by the computational search (Strategy A, sieve on d).

Key point: L = 41 is achieved by translation-based genealogy.

Notes

• Canonical setting: consecutive integers starting at n = 0, primality tested on |P(n)|.
• Stop rule: the first composite occurs at n = 41 for both polynomials.
• Historical remark: L = 41 matches Euler’s classical run length, now achieved within the monic quartic category.