login
Euler primes: Primes p that divide E(p - 3), where E(k) is the k-th Euler number.
6

%I #63 Sep 25 2023 09:43:02

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

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

%C Also called Vandiver primes. - _N. J. A. Sloane_, Sep 25 2023

%C See A196230 for another sequence of "Euler primes". - _N. J. A. Sloane_, May 29 2022

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

%C 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.

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

%C 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).

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

%C 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

%C If it exists, a(9) > 2 * 10^9. - _Hiroaki Yamanouchi_, Aug 06 2017

%C 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

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

%H A. R. Booker, S. Hathi, M. J. Mossinghoff and T. S. Trudgian, <a href="https://doi.org/10.1007/s11139-021-00438-3">Wolstenholme and Vandiver primes</a>, The Ramanujan Journal, 58 (2022), 913-941, arXiv:<a href="https://arxiv.org/abs/2101.11157">2101.11157</a>. See Theorem 1, but beware errors.

%H John B. Cosgrave and Karl Dilcher, <a href="https://doi.org/10.4064/aa161-1-4">On a congruence of Emma Lehmer related to Euler numbers</a>, Acta Arithmetica 161 (2013), 47-67.

%H Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, <a href="https://arxiv.org/abs/2101.11157">Wolstenholme and Vandiver primes</a>, arXiv:2101.11157 [math.NT], 2021.

%H R. J. McIntosh and E. L. Roettger, <a href="http://dx.doi.org/10.1090/S0025-5718-07-01955-2">A search for Fibonacci-Wieferich and Wolstenholme primes</a>, Math. Comp. vol 76, no 260 (2007) pp 2087-2094.

%H Romeo Mestrovic, <a href="http://arxiv.org/abs/1109.2340">An extension of a congruence by Kohnen</a>, arXiv: 1109.2340v3 [math.NT] (2011).

%H Romeo Mestrovic, <a href="http://arxiv.org/abs/1212.3602">A search for primes p such that Euler number E(p-3) is divisible by p</a>, arXiv: 1212.3602 [math.NT] (2012).

%H Zhi Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;62875835.1002">Letter to the Number Theory List</a>, Feb 8 2010

%H Zhi Wei Sun, <a href="http://arxiv.org/abs/1001.4453">Super congruences and Euler numbers</a>, Sci. China Math., 54 (2011), in press, arXiv: 1001.4453 [math.NT] (2011).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerNumber.html">Euler Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler_number">Euler Number</a>.

%H Hiroaki Yamanouchi, <a href="/A198245/a198245.txt">Primes p (5 <= p < 2*10^9) such that E(p-3) == A (mod p) for some integer A in [-1000, 1000].</a>

%t Select[Prime[Range[2, 200]], IntegerQ[EulerE[# - 3]/#] &] (* _Alonso del Arte_, Oct 31 2011 *)

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

%K nonn,more

%O 1,1

%A _Romeo Mestrovic_, Oct 22 2011

%E a(4)-a(8) from _Hiroaki Yamanouchi_, Aug 06 2017