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 A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4). 25
 16843, 2124679 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS McIntosh and Roettger showed that the next term, if it exists, must be larger than 10^9. - Felix Fröhlich, Aug 23 2014 When cb(m)=binomial(2m,m) denotes m-th central binomial coefficient then, obviously, cb(a(n))=2 mod a(n)^4. I have verified that among all naturals 1 1 for all i > 0. Whether this is true for all such n is an open problem. - Felix Fröhlich, Feb 21 2018 Primes p such that binomial(2*p-1, p-1) == 1-2*p*Sum_{k=1..p-1} 1/k - 2*p^2*Sum_{k=1..p-1} 1/k^2 (mod p^7) (cf. Mestrovic, 2011, Corollary 4). - Felix Fröhlich, Feb 21 2018 These are primes p such that p^2 divides A007406(p-1) (Mestrovic, 2015, p. 241, Lemma 2.3). - Amiram Eldar and Thomas Ordowski, Jul 29 2019 REFERENCES R. K. Guy, Unsolved Problems in Number Theory, Sect. B31. Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (Springer, 1979). LINKS J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp., 61 (1993), 151-153. Chris Caldwell, The Prime Glossary, Wolstenholme prime K. Conrad, The p-adic growth of harmonic sums R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica 71 (4): 381-389, (1995), 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. R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011. R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4. Romeo Mestrovic, Congruences for Wolstenholme Primes, arXiv:1108.4178 [math.NT], 2011. Romeo Mestrovic, Congruences for Wolstenholme Primes, Czechoslovak Mathematical Journal, 65 (2015), 237-253. Romeo Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012. Romeo Meštrović, Several generalizations and variations of Chu-Vandermonde identity, arXiv:1807.10604 [math.CO], 2018. J. Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; arXiv:1812.07650 [math.NT], 2018. Eric Weisstein's World of Mathematics, Wolstenholme Prime Eric Weisstein's World of Mathematics, Integer Sequence Primes Wikipedia, Wolstenholme prime J. Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, J. Number Theory, 123 (2007), 18-26. FORMULA A000984(a(n)) = 2 mod a(n)^4. - Stanislav Sykora, Aug 26 2014 A099908(a(n)) == 1 mod a(n)^4. - Jonathan Sondow, Nov 24 2015 A034602(PrimePi(a(n))) == 0 mod a(n) and A263882(PrimePi(a(n))) == 0 mod a(n)^2. - Jonathan Sondow, Dec 03 2015 PROG (PARI) forprime(n=2, 10^9, if(Mod(binomial(2*n-1, n-1), n^4)==1, print1(n, ", "))); \\ Felix Fröhlich, May 18 2014 (MAGMA) [p: p in PrimesUpTo(2*10^4)| (Binomial(2*p-1, p-1) mod (p^4)eq 1)]; // Vincenzo Librandi, May 02 2015 CROSSREFS Cf. A000984, A001008, A007406, A027641, A034602, A099908, A246130, A246132, A246133, A246134, A263882, A267824, A298946. Sequence in context: A237806 A061364 A203891 * A234699 A204639 A233986 Adjacent sequences:  A088161 A088162 A088163 * A088165 A088166 A088167 KEYWORD hard,nonn,bref,more AUTHOR Christian Schroeder, Sep 21 2003 STATUS approved

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Last modified October 15 22:25 EDT 2019. Contains 328038 sequences. (Running on oeis4.)