Long Prime Chains Generated by Monic Polynomials of Degrees 3 to 6

We report four monic integer polynomials, of degrees three to six, that generate long chains of prime absolute values. The cubic \(P_3(n)=n^3+54n^2-2917n+20021\) generates 40 consecutive prime values for \(n=0,\ldots,39\), containing 39 distinct primes. The quartic \(P_4(n)=n^4-2n^3-460n^2+461n+52027\), symmetric about \(x=1/2\), generates 68 consecutive prime values for \(n=-33,\ldots,34\), containing 34 distinct primes; its standardization \(Q_4(n)=P_4(n-33)\) takes prime values for \(n=0,\ldots,67\) and is symmetric about \(n=67/2\). The quintic \(P_5(n)=n^5+5n^4-29n^3-81n^2+30n+157\) generates 34 consecutive prime values for \(n=-20,\ldots,13\), containing 34 distinct primes; its standardization \(Q_5(n)=P_5(n-20)\) takes prime values for \(n=0,\ldots,33\). The sextic \(P_6(n)=n^6-3n^5+911n^4-1817n^3-3905n^2+4813n+5903\), symmetric about \(x=1/2\), generates 54 consecutive prime values for \(n=-26,\ldots,27\), containing 27 distinct primes; its standardization \(Q_6(n)=P_6(n-26)\) takes prime values for \(n=0,\ldots,53\) and is symmetric about \(n=53/2\). All primality tests are performed on absolute values. The quartic chain is the second-longest known chain for monic integer polynomials of any degree, after the classical quadratic record of 80. The sextic and quintic chains are, to the author's knowledge, the longest known within their respective degrees. We describe a point-value enumeration method, inspired by both the small-value condition and genealogical dynamics, which is more efficient than coefficient enumeration and makes these searches computationally feasible.