This monic sextic polynomial generates 30 consecutive prime values in canonical form,
i.e. for n = 0,1,…,29. The run stops because P(30) is composite.
Termination
P(30) = 10007743 = 1091 × 9173 (composite).
Remark on centered structure
The central part of the sequence is strikingly compressed: near the middle, the values become very small
and display an almost symmetric pattern
643, 727, 863, 883, 907, 863, 727.
The 30 consecutive prime values generated by P(n) in canonical form:
Since P(-1) is composite, this polynomial has no effective child
and is therefore a genealogical peak in the canonical monic setting.
P(-1) = 16941203 = 107 × 158329
The full unique ancestor chain is obtained by repeated shifts n → n + 1.
The table below lists the entire genealogical tree on this line, from the root L = 1
up to the peak L = 30.
| L | Monic sextic polynomial |
|---|---|
| 1 | n^6 + 83n^5 + 2857n^4 + 52201n^3 + 533904n^2 + 2897874n + 6520823 |
| 2 | n^6 + 77n^5 + 2457n^4 + 41583n^3 + 393628n^2 + 1975650n + 4107427 |
| 3 | n^6 + 71n^5 + 2087n^4 + 32505n^3 + 282866n^2 + 1303694n + 2486203 |
| 4 | n^6 + 65n^5 + 1747n^4 + 24847n^3 + 196652n^2 + 831586n + 1434887 |
| 5 | n^6 + 59n^5 + 1437n^4 + 18489n^3 + 130580n^2 + 505164n + 781423 |
| 6 | n^6 + 53n^5 + 1157n^4 + 13311n^3 + 81276n^2 + 290954n + 395803 |
| 7 | n^6 + 47n^5 + 907n^4 + 9183n^3 + 47188n^2 + 157538n + 182627 |
| 8 | n^6 + 41n^5 + 687n^4 + 6027n^3 + 25184n^2 + 78810n + 74383 |
| 9 | n^6 + 35n^5 + 497n^4 + 3717n^3 + 12354n^2 + 34506n + 25447 |
| 10 | n^6 + 29n^5 + 337n^4 + 2127n^3 + 5412n^2 + 11418n + 6803 |
| 11 | n^6 + 23n^5 + 207n^4 + 1131n^3 + 1902n^2 + 2418n + 1483 |
| 12 | n^6 + 17n^5 + 107n^4 + 603n^3 + 382n^2 + 162n + 727 |
| 13 | n^6 + 11n^5 + 37n^4 + 287n^3 + 82n^2 + 126n + 863 |
| 14 | n^6 + 5n^5 - 3n^4 + 153n^3 + 132n^2 + 319n + 907 |
| 15 | n^6 - n^5 - 5n^4 + 5n^3 - 3n^2 + 307n + 883 |
| 16 | n^6 - 7n^5 + 7n^4 + 99n^3 - 95n^2 - 31n + 863 |
| 17 | n^6 - 13n^5 + 57n^4 - n^3 - 17n^2 + 151n + 727 |
| 18 | n^6 - 19n^5 + 137n^4 - 635n^3 + 551n^2 - 859n + 643 |
| 19 | n^6 - 25n^5 + 247n^4 - 1607n^3 + 2374n^2 - 3533n + 2267 |
| 20 | n^6 - 31n^5 + 387n^4 - 3043n^3 + 6607n^2 - 14061n + 10663 |
| 21 | n^6 - 37n^5 + 557n^4 - 5081n^3 + 13324n^2 - 31683n + 36943 |
| 22 | n^6 - 43n^5 + 757n^4 - 7873n^3 + 24547n^2 - 68195n + 101627 |
| 23 | n^6 - 49n^5 + 987n^4 - 11583n^3 + 42164n^2 - 135399n + 238723 |
| 24 | n^6 - 55n^5 + 1247n^4 - 16377n^3 + 68539n^2 - 253863n + 500527 |
| 25 | n^6 - 61n^5 + 1537n^4 - 22423n^3 + 106488n^2 - 454092n + 963143 |
| 26 | n^6 - 67n^5 + 1857n^4 - 29889n^3 + 159380n^2 - 780156n + 1732723 |
| 27 | n^6 - 73n^5 + 2207n^4 - 38943n^3 + 230407n^2 - 1292887n + 2952427 |
| 28 | n^6 - 79n^5 + 2587n^4 - 49753n^3 + 323356n^2 - 2078270n + 4810103 |
| 29 | n^6 - 85n^5 + 2997n^4 - 62487n^3 + 442820n^2 - 3271422n + 7546687 |
| 30 | n^6 - 91n^5 + 3437n^4 - 68961n^3 + 775184n^2 - 4628206n + 11465323 |
PARI/GP
P(n)=n^6-91*n^5+3437*n^4-68961*n^3+775184*n^2-4628206*n+11465323;
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)));
SageMath / SageCell
def P(n):
return n^6 - 91*n^5 + 3437*n^4 - 68961*n^3 + 775184*n^2 - 4628206*n + 11465323
for n in range(0, 200):
v = P(n)
if not is_prime(v):
print("STOP n =", n, " P(n) =", v, " factor =", factor(v))
break
print("P(-1) =", P(-1), " factor =", factor(P(-1)))