A298946 -id:A298946 - cp's OEIS frontend

A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).

16843, 2124679

Offset: 1

Christian Schroeder, Sep 21 2003

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 1A246134). One might therefore wonder whether this is true in general. - Stanislav Sykora, Aug 26 2014
Romeo Mestrovic, Congruences for Wolstenholme Primes, Lemma 2.3, shows that the criterion for p to be a Wolstenholme prime is equivalent to p dividing A027641(p-3). In 1847 Cauchy proved that this was a necessary condition for the failure of the first case of Fermat's Last Theorem for the exponent p (see Ribenboim, 13 Lectures, p. 29). - John Blythe Dobson, May 01 2015
Primes p such that p^3 divides A001008(p-1) (Zhao, 2007, p. 18). Also: Primes p such that (p, p-3) is an irregular pair (cf. Buhler, Crandall, Ernvall, Metsänkylä, 1993, p. 152). Keith Conrad observes that for the two known (as of 2015) terms ord_p(H_p-1) = 3 is satisfied, where ord_p(H_p-1) gives the p-adic valuation of H_p-1 (cf. Conrad, p. 5). Romeo Mestrovic conjectures that p is a Wolstenholme prime if and only if S_(p-2)(p) == 0 (mod p^3), where S_k(i) denotes the sum of the k-th powers of the positive integers up to and including (i-1) (cf. Mestrovic, 2012, conjecture 2.10). - Felix Fröhlich, May 20 2015
Primes p that divide the Wolstenholme quotient W_p (A034602). Also, primes p such that p^2 divides the Babbage quotient b_p (A263882). - Jonathan Sondow, Nov 24 2015
The only known composite numbers n such that binomial(2n-1, n-1) is congruent to 1 mod n^2 are the numbers n = p^2 where p is a Wolstenholme prime: see A267824. - Jonathan Sondow, Jan 27 2016
The converse of Wolstenholme's theorem implies that if an integer n satisfies the congruence binomial(2*n-1, n-1) == 1 (mod n^4), then n is a term of this sequence, i.e., then n is necessarily prime, or, equivalently, A298946(i) > 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
If a third Wolstenholme prime exists it is larger than 6*10^10 (cf. Hathi, Mossinghoff, Trudgian, 2021). - Felix Fröhlich, Apr 27 2021
Named after the English mathematician Joseph Wolstenholme (1829-1891). - Amiram Eldar, Jun 10 2021
Primes p such that tanh(Sum_{k=1..p-1} artanh(k/p)) == 0 (mod p^4). - Thomas Ordowski, Apr 17 2025

Richard K. Guy, Unsolved Problems in Number Theory, Sect. B31.
Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (Springer, 1979).
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 23.

Ronald Bruck, Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients.
Joe Buhler, Richard Crandall, Reijo Ernvall and Tauno Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp., Vol. 61, No. 203 (1993), pp. 151-153.
Chris Caldwell, The Prime Glossary, Wolstenholme prime.
Leonardo Carofiglio, Luigi De Filpo, and Alessandro Gambini, p-adic valuation of harmonic sums and their connections with Wolstenholme primes, arXiv:2303.15010 [math.NT], 2023.
Keith Conrad, The p-adic growth of harmonic sums.
Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, Wolstenholme and Vandiver primes, The Ramanujan Journal, (2021); arXiv version, 2101.11157 [math.NT], 2021.
Richard J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica, Vol. 71, No. 4 (1995), pp. 381-389.
Richard J. McIntosh and Eric 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.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4.
Romeo Meštrović, Congruences for Wolstenholme Primes, arXiv:1108.4178 [math.NT], 2011.
Romeo Meštrović, Congruences for Wolstenholme Primes, Czechoslovak Mathematical Journal, Vol. 65 (2015), pp. 237-253.
Romeo Meštrović, 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.
Jonathan Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, CANT 2015 and 2016, New York, 2017, pp. 269-277; 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.
Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, J. Number Theory, Vol. 123 (2007), pp. 18-26.

Cf. A000984, A001008, A007406, A027641, A034602, A099908, A246130, A246132, A246133, A246134, A263882, A267824, A298946.

[p: p in PrimesUpTo(2*10^4)| (Binomial(2*p-1,p-1) mod (p^4)eq 1)]; // Vincenzo Librandi, May 02 2015

For[i = 2, i <= 20000, i++, {If[PrimeQ[i] && Mod[Binomial[2*i - 1, i - 1], i^4] == 1, Print[i]]}] (* Dylan Delgado, Mar 02 2021 *)

forprime(n=2, 10^9, if(Mod(binomial(2*n-1, n-1), n^4)==1, print1(n, ", "))); \\ Felix Fröhlich, May 18 2014

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

A298944 a(n) = 2^(c-1) mod c^2, where c is the n-th composite number.

Original entry on oeis.org

8, 32, 0, 13, 12, 32, 156, 184, 0, 176, 288, 319, 464, 320, 341, 496, 40, 64, 212, 0, 301, 308, 9, 1040, 952, 472, 1088, 1544, 800, 391, 508, 2048, 1191, 1312, 922, 2608, 284, 2359, 1920, 688, 1800, 3488, 2668, 2524, 0, 2291, 428, 144, 3109, 2612, 1472, 2888

Offset: 1

Felix Fröhlich, Jan 30 2018

nonn

a(n) = 0 iff c is a term of A000079 > 4.
Composites c where a(n) = 1 could be called "Wieferich pseudoprimes". Do any such composites exist?
A necessary condition for c to be a "Wieferich pseudoprime" would be that it is a term of both A001567 and A270833 (see comments in A240719).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000079, A001220, A001567, A240719, A270833, A298945, A298946.

map(c -> 2&^(c-1) mod c^2, remove(isprime, [$4..1000])); # Robert Israel, Feb 27 2018

composite[n_Integer] := FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1]; Array[With[{c = composite@ #}, Mod[2^(c - 1), c^2]] &, 52] (* Michael De Vlieger, Jan 31 2018, composite function by Robert G. Wilson v at A066277 *)

forcomposite(c=1, 200, print1(lift(Mod(2, c^2)^(c-1)), ", "))

A298945 a(n) = F_{c-(5/c)} mod c^2, where c is the n-th composite number, F_i = A000045(i) and (5/c) is the Kronecker symbol.

Original entry on oeis.org

2, 5, 34, 21, 55, 89, 37, 160, 98, 293, 365, 150, 101, 433, 25, 665, 696, 709, 440, 994, 883, 1090, 765, 1241, 230, 1511, 1355, 257, 805, 20, 1382, 289, 2275, 1525, 1414, 821, 1373, 1820, 685, 1504, 2177, 720, 3102, 1302, 1250, 190, 2425, 2178, 2832, 3935

Offset: 1

Felix Fröhlich, Jan 30 2018

nonn

Composites c where a(n) = 0 could be called "Wall-Sun-Sun pseudoprimes" or "Fibonacci-Wieferich pseudoprimes". Do any such composites exist?

Any such c would have to be a term of A241505.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A241505, A298944, A298946.

N:= 100: # to get a(1)..a(N)
count:= 0: R:= NULL:
for n from 4 while count < N do
if not isprime(n) then
  count:= count+1;
  R:= R, combinat:-fibonacci(n - numtheory:-jacobi(5,n)) mod n^2;
fi
od:
R; # Robert Israel, Feb 02 2018

composite[n_Integer] := FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1] ; Array[With[{c = composite@ #}, Mod[Fibonacci[c - KroneckerSymbol[5, c]], c^2]] &, 50] (* Michael De Vlieger, Jan 31 2018, composite function by Robert G. Wilson v at A066277 *)

forcomposite(c=1, 200, print1(lift(Mod(fibonacci(c-kronecker(5, c)), c^2)), ", "))

cp's OEIS Frontend

A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A298944 a(n) = 2^(c-1) mod c^2, where c is the n-th composite number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A298945 a(n) = F_{c-(5/c)} mod c^2, where c is the n-th composite number, F_i = A000045(i) and (5/c) is the Kronecker symbol.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI