Cubic Polynomial (L = 33) – Canonical Monic Peak

The Polynomial

This monic cubic polynomial generates 33 consecutive prime values in canonical form, for \(n=0,1,\ldots,32\). The run terminates at \(n=33\), where \(P(33)\) is composite.

Termination

\(P(33)=19979261=149\cdot134089\) (composite).

Genealogical peak

The value immediately preceding the canonical run is also composite:

\(P(-1)=3331807=173\cdot19259\).

Prime Sequence (n = 0..32)

The 33 consecutive prime values generated by P(n) are:

Compact form:

Form Found by the Search

The computational scan first identified the translated cubic line

with a run of 33 consecutive prime values for \(n=-2,-1,\ldots,30\). The translation \(P(n)=F(n-2)\) moves the run to the canonical interval \(n=0,\ldots,32\) and yields the polynomial above.

Computational Strategy

The final search was carried out on centered representatives of monic cubic genealogical lines of the form

First filter: modular obstructions for p ≤ 31

For a prime p, let N_p(F) denote the number of roots of F modulo p. The search imposed \(N_p(F)=0\) for every prime \(p\le31\). In the numerical domain considered, the presence of a root modulo one of these primes obstructs a block of 32 consecutive prime values, so this is a necessary condition for the target \(L\ge32\).

The 32-tuple filter

For each prime p, define

If \(\nu_{p,32}=p\), every possible starting residue class of a 32-value window is obstructed modulo p, and the line can be discarded without any primality test.

A cubic has at most three roots modulo p, and each root can obstruct at most 32 starting residue classes. Hence complete coverage can occur only for \(p\le96\). After the primes up to 31 have already been treated, it is sufficient to test the new complete obstructions for primes from 37 to 89.

Modular indicators

The general modular fertility indicator used in the search was

To evaluate specifically the configuration of 32 consecutive values, the search introduced

The second indicator was not used as a hard exclusion threshold. After the necessary filters and the broad condition \(S_{97}\ge6.30\), the surviving lines were ranked by \(T_{32,251}\) and tested in descending order. Thus the predictor determined exploration priority, not final admissibility.

Predictor calibration

Before the final scan, a calibration experiment was performed on 5,503 lines already compatible with a 32-tuple. In that sample, the Pearson correlations with the maximum run length observed in the central region were approximately:

In the top 1% of the distribution, \(T_{32,251}\) concentrated 23 cases with \(L\ge10\) and 3 cases with \(L\ge15\), compared with 17 and 1 respectively for \(S_{97}\). These results were used only as statistical evidence for search prioritization, not as a deterministic criterion.

Final scan

The final search used:

\(12001\le A\le18000\), \(1800001\le q\le3400000\) with q prime, candidate starts \(-256\le k\le256\), and an initial target of \(L=32\).

The search engine targeted a window of 32 prime values. The discovered run contains 33 values; consequently, two overlapping 32-value windows were recorded and later recognized as belonging to the same maximum run \(L=33\).

For the discovered line: \(S_{97}=7.360820004764\), \(\log T_{32,251}=65.463520889549\).

Translation Genealogy

The unique ancestor chain is obtained by repeated translations \(n\mapsto n+1\). The table below gives the complete genealogy from the root \(L=1\) to the peak \(L=33\).

Reproducibility

PARI/GP

P(n)=n^3+17375*n^2-67426*n+3247007;

for(n=0,200,
  if(!isprime(P(n)),
    print("STOP n=",n," P(n)=",P(n)," fact=",factor(P(n)));
    break
  )
);

print("P(-1)=",P(-1)," fact=",factor(P(-1)));

Expected output:

STOP n=33 P(n)=19979261 fact=[149,1; 134089,1]
P(-1)=3331807 fact=[173,1; 19259,1]

Summary

Note on Optimality

Zenodo

Related Work

• Genealogy Theorem for Monic Prime-Generating Polynomials
• Genealogical Dynamics of Monic Prime-Generating Polynomials
• A monic sextic polynomial with L = 36