A245206 Odd primes p with E_{p-3}(1/4) == 0 (mod p), where E_n(x) denotes the Euler polynomial of degree n.

1019

Offset: 1

Author

Zhi-Wei Sun, Jul 13 2014

Keywords

• nonn
• more
• hard
• bref

Comments

The conjecture in A245204 asserts that the current sequence contains infinitely many primes.

Our computation shows that the second term should be greater than prime(2600) = 23321.

Examples

			a(1) = 1019 since 1019 is a prime with E_{1019-3}(1/4) == 88*1019 (mod 1019^2).
		

Links

• Zhi-Wei Sun, Super congruences and Euler numbers, Sci. China Math., Vol. 54, No. 12 (2011), 2509-2535; arXiv preprint, arXiv:1001.4453 [math.NT], 2010-2011.
• Index entries for one-term sequences.

Crossrefs

Cf. A000040, A001586, A122045, A198245, A245089, A245204.

Programs

Mathematica

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
n=0;Do[If[rMod[EulerE[Prime[k]-3,1/4],Prime[k]]==0,n=n+1;Print[n," ",Prime[k]]],{k,2,200}]