A165284 Primes p in A068209 whose squares never divide (x+1)^p-x^p-1 and x^x+(x+1)^(x+1) for the same x

37493, 51941, 58073, 58901, 83813, 252341, 278321, 366521, 369821, 375101, 405689, 461861, 611801, 647837, 739061, 832721, 902201, 1001081, 1102301, 1180961, 1220801, 1269041, 1283297, 1426361, 1448081, 1483637, 1486577

Offset

1,1

Comments

A prime p belongs to A068209 if and only if p = 5 mod 6 and there are integers x with (x+1)^p - x^p - 1 = 0 mod p^2 and gcd(x^2+x,p) = 1.

This sequence is the subsequence of A068209 of primes p for which no such x solves x^x + (x+1)^(x+1) = 0 mod p^2.

For all other primes p < 1486577 in A068209, simultaneous solutions have been found by computing discrete logarithms.

Links

Table of n, a(n) for n=1..27.

David Broadhurst, On roots of n^n + (n+1)^(n+1) = 0 mod p^2

Kevin Brown, On the Density of Some Exceptional Primes

Example

To prove that a(3) = 58073, we first show that (x+1)^p - x^p - 1 mod p^2, with gcd(x^2+x,p) = 1, has solutions when p = 58073 only for the residues x = r, -r/(1+r), 1/r, -(1+r), -1/(1+r), -(1+1/r) mod p, with r = 1281. By examining the orders of 1+1/r, 1+r, -r mod p, we prove that no x in this equivalence class can satisfy x^x + (x+1)^(x+1) = 0 mod p^2.
Similarly, we prove the absence of simultaneous roots for p = 37493, with r = 3730, and for p = 51941, with r = 15579.
By computing discrete logarithms, we provide simultaneous solutions for all other primes in A068209 with p < 58073.

Crossrefs

Cf. A068209

Sequence in context: A215993 A022245 A236744 * A235298 A233662 A284080

Adjacent sequences: A165281 A165282 A165283 * A165285 A165286 A165287

Keyword

nonn

Author

David Broadhurst, Sep 13 2009

Status

approved