This monic sextic polynomial generates 36 consecutive prime values in canonical form,
i.e. for n = 0,1,…,35. The run stops because P(36) is composite.
Termination
P(36) = 115278749 = 701 × 164449 (composite).
Remark on structure
This sextic is a genuine genealogical peak, but unlike some symmetric quartic peaks, its prime sequence is not palindromic. This makes it especially interesting within the broader theory of polynomial genealogies.
The 36 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) = 10076279 = 167 × 60337
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 = 36.
| L | Monic sextic polynomial |
|---|---|
| 1 | n^6 + 127n^5 + 6707n^4 + 188513n^3 + 2973954n^2 + 24966960n + 87142487 |
| 2 | n^6 + 121n^5 + 6087n^4 + 162935n^3 + 2447402n^2 + 19558392n + 64967549 |
| 3 | n^6 + 115n^5 + 5497n^4 + 139777n^3 + 1993924n^2 + 15128644n + 47699591 |
| 4 | n^6 + 109n^5 + 4937n^4 + 118919n^3 + 1606440n^2 + 11538708n + 34430477 |
| 5 | n^6 + 103n^5 + 4407n^4 + 100241n^3 + 1278230n^2 + 8663376n + 24384119 |
| 6 | n^6 + 97n^5 + 3907n^4 + 83623n^3 + 1002934n^2 + 6390520n + 16903037 |
| 7 | n^6 + 91n^5 + 3437n^4 + 68945n^3 + 774552n^2 + 4620372n + 11435639 |
| 8 | n^6 + 85n^5 + 2997n^4 + 56087n^3 + 587444n^2 + 3264804n + 7524221 |
| 9 | n^6 + 79n^5 + 2587n^4 + 44929n^3 + 436330n^2 + 2246608n + 4793687 |
| 10 | n^6 + 73n^5 + 2207n^4 + 35351n^3 + 316290n^2 + 1498776n + 2940989 |
| 11 | n^6 + 67n^5 + 1857n^4 + 27233n^3 + 222764n^2 + 963550n + 1725287 |
| 12 | n^6 + 61n^5 + 1537n^4 + 20455n^3 + 151552n^2 + 594780n + 958829 |
| 13 | n^6 + 55n^5 + 1247n^4 + 14847n^3 + 98681n^2 + 348126n + 498551 |
| 14 | n^6 + 49n^5 + 987n^4 + 10281n^3 + 60644n^2 + 191704n + 238397 |
| 15 | n^6 + 43n^5 + 757n^4 + 6713n^3 + 34517n^2 + 97929n + 102359 |
| 16 | n^6 + 37n^5 + 557n^4 + 4097n^3 + 17774n^2 + 45373n + 38237 |
| 17 | n^6 + 31n^5 + 387n^4 + 2387n^3 + 7987n^2 + 17701n + 12119 |
| 18 | n^6 + 25n^5 + 247n^4 + 1297n^3 + 2854n^2 + 4453n + 3581 |
| 19 | n^6 + 19n^5 + 137n^4 + 501n^3 + 961n^2 + 501n + 1607 |
| 20 | n^6 + 13n^5 + 57n^4 + 121n^3 + 181n^2 + 49n + 1229 |
| 21 | n^6 + 7n^5 + 7n^4 + 41n^3 + 29n^2 + 73n + 887 |
| 22 | n^6 + n^5 - 13n^4 + 25n^3 + 17n^2 + 311n + 509 |
| 23 | n^6 - 5n^5 - 3n^4 + 65n^3 - 15n^2 + 191n + 311 |
| 24 | n^6 - 11n^5 + 37n^4 - 95n^3 + 107n^2 + 167n + 317 |
| 25 | n^6 - 17n^5 + 107n^4 - 603n^3 + 1217n^2 - 637n + 599 |
| 26 | n^6 - 23n^5 + 207n^4 - 1817n^3 + 4717n^2 - 6251n + 2237 |
| 27 | n^6 - 29n^5 + 337n^4 - 4047n^3 + 12767n^2 - 22511n + 8999 |
| 28 | n^6 - 35n^5 + 497n^4 - 7577n^3 + 27870n^2 - 57680n + 29741 |
| 29 | n^6 - 41n^5 + 687n^4 - 12681n^3 + 51934n^2 - 122973n + 81527 |
| 30 | n^6 - 47n^5 + 907n^4 - 19633n^3 + 85967n^2 - 227985n + 193469 |
| 31 | n^6 - 53n^5 + 1157n^4 - 28707n^3 + 129177n^2 - 373895n + 411287 |
| 32 | n^6 - 59n^5 + 1437n^4 - 40177n^3 + 182972n^2 - 586260n + 802589 |
| 33 | n^6 - 65n^5 + 1747n^4 - 54317n^3 + 248960n^2 - 874127n + 1462871 |
| 34 | n^6 - 71n^5 + 2087n^4 - 71303n^3 + 328949n^2 - 1252336n + 2522237 |
| 35 | n^6 - 77n^5 + 2457n^4 - 91311n^3 + 424947n^2 - 1733482n + 4152839 |
| 36 | n^6 - 83n^5 + 2857n^4 - 52217n^3 + 534664n^2 - 2909420n + 6577037 |
PARI/GP
P(n)=n^6-83*n^5+2857*n^4-52217*n^3+534664*n^2-2909420*n+6577037;
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 - 83*n^5 + 2857*n^4 - 52217*n^3 + 534664*n^2 - 2909420*n + 6577037
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)))