A342066 Primes p such that p^10 - 1 has 256 divisors.

1187, 4723, 33037, 66973, 72797, 87973, 100523, 197123, 219683, 229693, 276293, 278827, 440653, 448997, 482837, 562963, 601333, 621443, 670493, 742723, 846877, 892357, 1033427, 1149307, 1166027, 1245067, 1256747, 1614413, 1679773, 1865693, 1950323, 1970467

Offset

1,1

Comments

Conjecture: sequence is infinite.

The only primes p such that p^10 - 1 has fewer than A309906(10)=256 divisors are 2, 3, 5, 7, 11, 13, and 43.

p^10 - 1 = (p-1)*(p+1)*(p^4 - p^3 + p^2 - p + 1)*(p^4 + p^3 + p^2 + p + 1). For every p > 11, one of these five factors is divisible by 11; one of p-1 and p+1 is divisible by 3; and p-1 and p+1 are consecutive even numbers, so one of them is divisible by 4 and their product is divisible by 8; thus, p^10 - 1 is divisible by 2^3 * 3 * 11.

For every term p with the exception of a(1)=1187, p^10 - 1 is of the form 2^3 * 3 * 11 * q * r * s * t, where q, r, s, and t are distinct primes > 11.

Links

Table of n, a(n) for n=1..32.

Example

For p = a(1) = 1187, p^10 - 1 = 2^3 * 3^3 * 11 * 593 * 1983522604541 * 1986867499321;
for p = a(2) = 4723, p^10 - 1 = 2^3 * 3 * 11 * 787 * 1181 * 45245048697451 * 497484826300381.

Crossrefs

Cf. A000005, A000040, A309906, A341670.

Sequence in context: A237698 A345664 A052236 * A153378 A205230 A229539

Adjacent sequences: A342063 A342064 A342065 * A342067 A342068 A342069

Keyword

nonn

Author

Jon E. Schoenfield, Feb 27 2021

Status

approved