Monic quartic prime run with L = 49

The polynomial

Q(n) = n^4 − 96n^3 + 3153n^2 − 40752n + 192307

This monic quartic integer polynomial generates a sequence of 49 consecutive prime values when evaluated at n = 0, 1, …, 48. Primality is tested on the absolute value Q(n).

Run length: L = 49

Context

Euler’s famous polynomial n^2 + n + 41 produces primes for n = 0..39 (then n=40 gives 1681 = 41^2). In terms of consecutive primes starting at n=0, the monic quartic documented here reaches L = 49.

Translation genealogy (L = 42 → 49)

The record polynomial Q(n) is obtained through repeated translations of the form:

Q(n) = P(n − 1)

If P(n) is prime for n = 0..L-1 and P(-1) is also prime, then the translation increases the run length by one: the translated polynomial produces primes for n = 0..L. Iterating this mechanism produces a genealogical chain.

Starting polynomial (L = 42)

P(n) = n^4 − 68n^3 + 1431n^2 − 9350n + 31013

For this polynomial, primality holds for n = 0..41, and the stop occurs at n = 42: P(42) = 236309 = 67 × 3527. Moreover, the negative-side values P(−1), …, P(−7) are prime, while P(−8) is composite, enabling 7 consecutive translations and leading to L = 49.

Complete genealogical table

List of generated prime numbers

Computational verification

Verification is performed by evaluating the polynomial at consecutive integer values of n, testing primality on |P(n)| until the first composite occurs. Factoring the stop value makes the check transparent and reproducible.

SageMath / SageCell (online)

# Paste into https://sagecell.sagemath.org/

def Q(n):
    return n^4 - 96*n^3 + 3153*n^2 - 40752*n + 192307

for n in range(0, 200):
    v = abs(Q(n))
    if not is_prime(v):
        print("STOP n =", n, " Q(n) =", v, " factor =", factor(v))
        break

PARI/GP (online or local)

Q(n)=n^4-96*n^3+3153*n^2-40752*n+192307;
for(n=0,200,
  if(!isprime(abs(Q(n))),
    print("STOP n=",n,"  Q(n)=",abs(Q(n)),"  fact=",factor(abs(Q(n))));
    break
  )
);

Expected output: STOP n=49 Q(n)=236309 fact=67*3527.