A198245 Euler primes: Primes p that divide E(p - 3), where E(k) is the k-th Euler number.

149, 241, 2946901, 16467631, 17613227, 327784727, 426369739, 1062232319

Offset

1,1

Comments

Also called Vandiver primes. - N. J. A. Sloane, Sep 25 2023

See A196230 for another sequence of "Euler primes". - N. J. A. Sloane, May 29 2022

The even-indexed Euler numbers are A028296, the odd-indexed Euler numbers are all zero.

Numerous combinatorial congruences recently obtained by Z. W. Sun and by Z. H. Sun contain the Euler numbers E(p-3) with a prime p.

Only three primes less than 3 * 10^6 satisfy this condition (the current members of the sequence).

Such primes have been recently suggested by Z. W. Sun; namely, Sun found the first and the second such primes, 149 and 241, and used them to discover new congruences involving E(p - 3).

This is reported by Zhi Wei Sun on Feb 08 2010 and the third prime was found by Romeo Mestrovic (on Sep 26 2011).

Mestrovic (2012) computes that only three primes < 10^7 are in the sequence, but he conjectures that the sequence is infinite. - Jonathan Sondow, Dec 18 2012

If it exists, a(9) > 2 * 10^9. - Hiroaki Yamanouchi, Aug 06 2017

Hathi et al. give a(3) as 2124679 and claim that the terms 2124679, 16467631, 17613227 were reported in Cosgrave, Dilcher, 2013, but 2124679 does not appear in table 2 in that paper. How is 2124679 related to this sequence? Note that 2124679 is the second Wolstenholme prime (A088164). - Felix Fröhlich, Apr 27 2021

References

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.8.

Links

Table of n, a(n) for n=1..8.

A. R. Booker, S. Hathi, M. J. Mossinghoff and T. S. Trudgian, Wolstenholme and Vandiver primes, The Ramanujan Journal, 58 (2022), 913-941, arXiv:2101.11157. See Theorem 1, but beware errors.

John B. Cosgrave and Karl Dilcher, On a congruence of Emma Lehmer related to Euler numbers, Acta Arithmetica 161 (2013), 47-67.

Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, Wolstenholme and Vandiver primes, arXiv:2101.11157 [math.NT], 2021.

R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. vol 76, no 260 (2007) pp 2087-2094.

Romeo Mestrovic, An extension of a congruence by Kohnen, arXiv: 1109.2340v3 [math.NT] (2011).

Romeo Mestrovic, A search for primes p such that Euler number E(p-3) is divisible by p, arXiv: 1212.3602 [math.NT] (2012).

Zhi Wei Sun, Letter to the Number Theory List, Feb 8 2010

Zhi Wei Sun, Super congruences and Euler numbers, Sci. China Math., 54 (2011), in press, arXiv: 1001.4453 [math.NT] (2011).

Eric Weisstein's World of Mathematics, Euler Number.

Wikipedia, Euler Number.

Hiroaki Yamanouchi, Primes p (5 <= p < 2*10^9) such that E(p-3) == A (mod p) for some integer A in [-1000, 1000].

Mathematica

Select[Prime[Range[2, 200]], IntegerQ[EulerE[# - 3]/#] &] (* Alonso del Arte, Oct 31 2011 *)

Crossrefs

Cf. A000364, A088164, A092217, A092218, A120115, A120337, A196230.

Sequence in context: A144315 A267490 A365405 * A142687 A142825 A127592

Adjacent sequences: A198242 A198243 A198244 * A198246 A198247 A198248

Keyword

nonn,more

Author

Romeo Mestrovic, Oct 22 2011

Extensions

a(4)-a(8) from Hiroaki Yamanouchi, Aug 06 2017

Status

approved