A124122 -id:A124122 - cp's OEIS frontend

A124121 Primes q which are the lesser of a double Wieferich prime pair. (List of known values of q in increasing order, without multiplicity.)

2, 3, 5, 83, 911, 2903

Offset: 1

N. J. A. Sloane, following an email from Robert G. Wilson v, Nov 30 2006

Double Wieferich prime pairs are pairs of primes (p, q) such that q^(p-1) == 1 (mod p^2) and p^(q-1) == 1 (mod q^2). This sequence gives the primes q which are the lesser member (listed second) of such pairs, in increasing order, but without multiplicity:
For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5). In this sequence, 5 is only listed once, as a(3), and only the lesser value p=1645333507 is listed as A124122(3).
This is just the list of known pairs: there may be gaps, i.e. missing primes.

Yuri F. Bilu, Catalan's Conjecture, Seminaire Bourbaki, (2002-2003).
Michael Mossinghoff, Wieferich Prime Pairs, Barker Sequences, and Circulant Hadamard Matrices, as of Feb 12 2009.
Eric Weisstein's World of Mathematics, Double Wieferich Prime Pair
Wikipedia, Wieferich pair

See A124122 for values of p.

Cf. A196511, A196733.

/* The following (highly unoptimized) code misses the value q=5 (corresponding to a very large value of p) */
default(primelimit,1010000); forprime(q=1, default(primelimit), forprime(p=q+1, default(primelimit),  Mod(p, q^2)^(q-1)-1 & next; Mod(q, p^2)^(p-1)-1 || print1( q", ") || break))  \\ M. F. Hasler, Oct 08 2011

A244550 a(n) = first odd Wieferich prime to base a(n-1) for n > 1, with a(1) = 2.

Original entry on oeis.org

2, 1093, 5, 20771, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71, 3, 11, 71

Offset: 1

Felix Fröhlich, Jun 29 2014

a(2) = 1093 since 1093 is the smallest odd Wieferich prime to base 2.
a(3) = 5 since 5 is the smallest odd Wieferich prime to base 1093.
Subsequence starting at a(5) is periodic with period 3, repeating the terms {3, 11, 71}.
Do values for a(1) exist such that the resulting sequence does not eventually become periodic?
The following table lists the values for a(1) and the resulting cycles those values produce. An entry of the form x-y in first column means all terms from x up to and including y reach the corresponding cycle. An entry of the form {t_1, t_2, t_3, ..., t_n} in second column means the listed terms form a repeating cycle. Entries in second column without curly braces mean the listed terms are reached in order and the term following the last listed term is unknown. A question mark means no further terms have been found in the resulting trajectory of a(1).
a(1)             | resulting terms
----------------------------------
2-13, 15-20,     | {3, 11, 71}
22-28, 30-40,    |
42-46, 48-59,    |
62-71, 73-82,    |
84-87, 89-118,   |
120-132, 134-136,|
138, 140-155,    |
157-185, 188,    |
190-195, 197-199 |
                 |
14, 41, 60, 137, | 29
196              |
                 |
21, 29, 47, 61,  | ?
72, 139, 186-187 |
                 |
83               | {4871, 83}
                 |
88               | 2535619637, 139
                 |
119              | 1741
                 |
133              | 5277179
                 |
156              | 347
                 |
189              | 1847
                 |
Notes
------
The terms of the cycle reached from 83 correspond to A124121(4) and A124122(4), so those terms form a double Wieferich prime pair.

R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A001220, A124121, A124122, A174422, A244546.

[2, 1093, 5, 20771] cat &cat [[3, 11, 71]^^30]; // Wesley Ivan Hurt, Jun 30 2016

2,1093,5,20771,seq(op([3, 11, 71]), n=5..50); # Wesley Ivan Hurt, Jun 30 2016

Join[{2, 1093, 5, 20771},LinearRecurrence[{0, 0, 1},{3, 11, 71},66]] (* Ray Chandler, Aug 25 2015 *)

i=0; a=2; print1(a, ", "); while(i<100, forprime(p=2, 10^6, if(Mod(a, p^2)^(p-1)==1 && p%2!=0, print1(p, ", "); i++; a=p; break({n=1}))))

From Wesley Ivan Hurt, Jun 30 2016: (Start)
G.f.: x*(2+1093*x+5*x^2+20769*x^3-1090*x^4+6*x^5-20700*x^6) / (1-x^3).
a(n) = a(n-3) for n>7.
a(n) = (85 - 52*cos(2*n*Pi/3) + 68*sqrt(3)*sin(2*n*Pi/3))/3 for n>4. (End)

A126432 Double Wieferich primes with q = 5.

Original entry on oeis.org

1645333507, 188748146801

Offset: 1

Alexander Adamchuk, Mar 12 2007

Double Wieferich prime pairs are pairs of primes (p, q) such that q^(p-1) == 1 (mod p^2) and p^(q-1) == 1 (mod q^2).

C. K. Caldwell, The Prime Glossary, Fermat quotient.
Wikipedia, Wieferich pair

Subsequence of A123692.

Cf. A124121, A124122, A125775.

A266829 Primes p such that a prime q < p exists with p^(q-1) == 1 (mod q^2) and q^(p-1) == 1 (mod p^2), i.e., primes that are the larger member of a double Wieferich prime pair.

Original entry on oeis.org

1093, 4871, 18787, 318917, 1006003, 1645333507

Offset: 1

Felix Fröhlich, Jan 04 2016

There are no further terms less than 10^6 (cf. Ernvall, Metsänkylä, 1997, p. 1360).
There are no further terms p less than 2^(1/3)*10^10 with p*q <= 10^15 and p and q both odd. (cf. Logan, Mossinghoff, results 4.2.). - Felix Fröhlich, May 29 2016 [Corrected. Felix Fröhlich, Aug 05 2018]
Primes that occur in column 2 of A282293. - Felix Fröhlich, Aug 05 2018

R. Ernvall and T. Metsänkylä, On the p-divisibility of Fermat quotients, Math. Comp., Volume 66, Number 219 (1997), 1353-1365.
B. Logan and M. J. Mossinghoff, Double Wieferich pairs and circulant Hadamard matrices, ResearchGate, 2015.

Cf. A124122, A282293, A317724 (smallest existing q).

fQ[p_] := Block[{q = 2}, While[q < p && (PowerMod[p, q - 1, q^2] != 1 || PowerMod[q, p - 1, p^2] != 1), q = NextPrime@ q]; If[q < p, True, False]]; p = 3; lst = {}; While[p < 1000000, If[fQ@ p, AppendTo[lst, p]]; p = NextPrime@ p]; lst (* Robert G. Wilson v, Jan 04 2016 *)

forprime(p=3, , forprime(q=2, p-1, if(Mod(p, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1, print1(p, ", "); break({1}))))

a(5)-a(6) from Felix Fröhlich, May 29 2016

Removed three comments. - Felix Fröhlich, Aug 21 2018

A196511 Primes q for which there are primes b < c < p such that b^p == c^p == 1 (mod q^2).

Original entry on oeis.org

555383, 1767407, 2103107, 2452757, 7400567, 12836987, 14668163, 15404867, 16238303, 19572647, 22796069, 25003799, 26978663, 27370727

Offset: 1

M. F. Hasler, Oct 03 2011

Computed mainly by M. Oakes and D. Broadhurst, cf. link.
For all the listed terms, except for a(4)=2452757 and a(11)=22796069, p=(q-1)/2, which implies that they are "safe primes", cf. A005385. For a(4) and a(11), p=(q-1)/4.
The subsequence of terms of the form q=2p+1 is A196733. As of today it is complete up to its 51st term, ~2e8. - M. F. Hasler, Oct 05 2011

D. J. Broadhurst et al., Re: Square factors of b^p-1 on yahoo group "primenumbers", Sept.-Oct. 2011
David Broadhurst and others, Square factors of b^p-1, digest of 81 messages in primenumbers Yahoo group, Sep 22 - Nov 29, 2011.

Cf. A196733, A124121, A124122, A005384, A005385.

A196733 Primes q = 2*p+1 for which there are primes b < c < p such that b^p == c^p == 1 (mod q^2).

Original entry on oeis.org

555383, 1767407, 2103107, 7400567, 12836987, 14668163, 15404867, 16238303, 19572647, 25003799, 26978663, 27370727, 35182919, 36180527, 38553023, 39714083, 52503587, 53061143, 53735699, 55072427, 63302159, 70728839, 77199743, 77401679, 86334299, 97298759, 97375319

Offset: 1

M. F. Hasler, Oct 05 2011

nonn

From D. Broadhurst, Oct 05 2011: (Start)
(p,q) is a Sophie Germain prime pair; (b,q) and (c,q) are Wieferich prime pairs; each of (b,c) is a square modulo q^2.
The sequence is now complete up to the 51st term, q=199065467.
It is a subsequence of A196511, where the latter does not require that q=2*p+1, is complete only up q=27370727, and contains q=2452757 and q=22796069, with q=4*p+1, (cf. link to post on "primenumbers" group), found by a simple analysis of Mossinghoff's results on Wieferich primes (cf. link).
With thanks to Mike Oakes. (End)

D. J. Broadhurst, Table of n, a(n) for n = 1..51
D. J. Broadhurst et al., Re: Square factors of b^p-1 on yahoo group "primenumbers", Sept.-Oct. 2011
David Broadhurst and others, Square factors of b^p-1, digest of 81 messages in primenumbers Yahoo group, Sep 22 - Nov 29, 2011.
Michael Mossinghoff, Wieferich Prime Pairs, Barker Sequences, and Circulant Hadamard Matrices, as of Feb 12 2009.

Cf. A005384, A005385, A124121, A124122, A196511.

A253683 Primes p in increasing order with p > A253684(n) > A253685(n) such that (p, A253684(n), A253685(n)) forms a Wieferich triple.

Original entry on oeis.org

71, 863, 1093, 2281, 3511, 13691, 20771, 54787, 950507, 1843757, 3188089

Offset: 1

Felix Fröhlich, Jan 09 2015

In analogy to a Wieferich pair, a set of three primes p, q, r can be called a 'Wieferich triple' if its members satisfy either of the following two sets of congruences:
p^(q-1) == 1 (mod q^2), q^(r-1) == 1 (mod r^2), r^(p-1) == 1 (mod p^2)
p^(r-1) == 1 (mod r^2), r^(q-1) == 1 (mod q^2), q^(p-1) == 1 (mod p^2)
a(9) > 121637. - Felix Fröhlich, Jun 18 2016
a(12) > 5*10^6. - Giovanni Resta, Jun 20 2016

Wikipedia, Wieferich pair

Cf. A124121, A124122, A253684, A253685.

forprime(p=1, , forprime(q=1, p, forprime(r=1, q, if((Mod(p, q^2)^(q-1)==1 && Mod(q, r^2)^(r-1)==1 && Mod(r, p^2)^(p-1)==1) || (Mod(p, r^2)^(r-1)==1 && Mod(r, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1), print1(p, ", ")))))

a(8) from Felix Fröhlich, Jun 18 2016
Name edited by Felix Fröhlich, Jun 18 2016
a(9)-a(11) from Giovanni Resta, Jun 20 2016

A271100 Triangular array read by rows: T(n, k) = k-th largest member of lexicographically earliest Wieferich n-tuple that contains no duplicate members, read by rows, or T(n, k) = 0 if no Wieferich n-tuple exists.

Original entry on oeis.org

0, 1093, 2, 71, 11, 3, 3511, 19, 13, 2, 359, 331, 71, 11, 3, 359, 331, 307, 71, 11, 3, 359, 331, 307, 71, 19, 11, 3, 863, 359, 331, 71, 23, 13, 11, 3, 863, 359, 331, 307, 71, 19, 13, 11, 3, 863, 467, 359, 331, 307, 71, 19, 13, 11, 3

Offset: 1

Felix Fröhlich, Mar 30 2016

Let p_1, p_2, p_3, ..., p_u be a set P of distinct prime numbers and let m_1, m_2, m_3, ..., m_u be a set V of variables. Then P is a Wieferich u-tuple if there exists a mapping from the elements of P to the elements of V such that each of the following congruences is satisfied:
m_1^(m_2-1) == 1 (mod (m_2)^2), m_2^(m_3-1) == 1 (mod (m_3)^2), ..., m_u^(m_1-1) == 1 (mod (m_1)^2).
For finding candidate values for m_1 given some m_u, one checks primes higher than m_u for primes satisfying m_u^(m_1-1) == 1 (mod (m_1)^2). For example, to see what we could get if m_u = 2, we check up to say m_1 = 1,000,000 to get candidates for m_1. This would give m_1 in {1093, 3511}. - David A. Corneth, May 14 2016

			For n = 1: There is no Wieferich singleton (1-tuple), because no prime p satisfies the congruence p^(p-1) == 1 (mod p^2), so T(1, 1) = 0.
For n = 4: The primes 3511, 19, 13, 2 satisfy the congruences 3511^(19-1) == 1 (mod 19^2), 19^(13-1) == 1 (mod 13^2), 13^(2-1) == 1 (mod 2^2) and 2^(3511-1) == 1 (mod 3511^2) and thus form a "Wieferich quadruple". Since this is the lexicographically earliest such set of primes, T(4, 1..4) = 3511, 19, 13, 2.
Triangle starts:
  n=1:     0;
  n=2:  1093,   2;
  n=3:    71,  11,   3;
  n=4:  3511,  19,  13,   2;
  n=5:   359, 331,  71,  11,   3;
  n=6:   359, 331, 307,  71,  11,   3;
  n=7:   359, 331, 307,  71,  19,  11,   3;
  n=8:   863, 359, 331,  71,  23,  13,  11,   3;
  n=9:   863, 359, 331, 307,  71,  19,  13,  11,   3;
  n=10:  863, 467, 359, 331, 307,  71,  19,  13,  11,   3;
  ....

Bruce Leenstra, Table of n, a(n) for n = 1..300
Wikipedia, Wieferich pair

Cf. A124121, A124122, A253683, A253684, A253685, A266829.

ulimupto(u,{llim=2}) = {my(l=List());
forprime(i=nextprime(llim+1),u,if(Mod(llim,i^2)^(i-1)==1,listput(l,i)));l} \\ David A. Corneth, May 14 2016
\\tests if a tuple is a valid Wieferich n-tuple.

istuple(v) = {if(#Set(v)==#v,return(0));for(j=0,(#v-1)!-1, w=vector(#v,k,v[numtoperm(#v,j)[k]]); if(sum(i=2,#w,Mod(w[i-1],w[i]^2)^(w[i]-1)==1)+(Mod(w[1],w[#w])^(w[#w]-1)==1)==#w,return(1)));0} \\ David A. Corneth, May 15 2016

wief = DiGraph([prime_range(3600), lambda p, q: power_mod(p, q-1, q^2)==1])
sc = [[0]] + [sorted(c[1:], reverse=1) for c in wief.all_simple_cycles()]
tbl = [sorted(filter(lambda c: len(c)==n, sc))[0] for n in range(1, len(sc[-1]))]
for t in tbl: print('n=%d:'% len(t), ', '.join("%s"%i for i in t)) # Bruce Leenstra, May 18 2016

a(11)-a(15) from Felix Fröhlich, Apr 26 2016

More terms from Bruce Leenstra, May 18 2016

A253684 Primes q with A253683(n) > q > A253685(n) such that (A253683(n), q, A253685(n)) forms a Wieferich triple.

Original entry on oeis.org

11, 23, 5, 1667, 73, 821, 18043, 2393, 20771, 2251, 1006003

Offset: 1

Felix Fröhlich, Jan 09 2015

In analogy to a Wieferich pair, a set of three primes p, q, r can be called a 'Wieferich triple' if its members satisfy either of the following two sets of congruences:
p^(q-1) == 1 (mod q^2), q^(r-1) == 1 (mod r^2), r^(p-1) == 1 (mod p^2)
p^(r-1) == 1 (mod r^2), r^(q-1) == 1 (mod q^2), q^(p-1) == 1 (mod p^2)
a(9) must have A253683(n) > 121637. - Felix Fröhlich, Jun 18 2016
a(12) must have A253683(n) > 5*10^6. - Giovanni Resta, Jun 20 2016

Wikipedia, Wieferich pair

Cf. A124121, A124122.

Cf. A253683, A253685.

forprime(p=1, , forprime(q=1, p, forprime(r=1, q, if((Mod(p, q^2)^(q-1)==1 && Mod(q, r^2)^(r-1)==1 && Mod(r, p^2)^(p-1)==1) || (Mod(p, r^2)^(r-1)==1 && Mod(r, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1), print1(q, ", ")))))

a(8) from Felix Fröhlich, Jun 18 2016
Name edited by Felix Fröhlich, Jun 18 2016
a(9)-a(11) from Giovanni Resta, Jun 20 2016

A253685 Primes r with A253683(n) > A253684(n) > r such that (A253683(n), A253684(n), r) is a Wieferich triple.

Original entry on oeis.org

3, 13, 2, 1657, 2, 83, 5, 431, 5, 199, 3

Offset: 1

Felix Fröhlich, Jan 09 2015

In analogy to a Wieferich pair, a set of three primes p, q, r can be called a 'Wieferich triple' if its members satisfy either of the following two sets of congruences:
p^(q-1) == 1 (mod q^2), q^(r-1) == 1 (mod r^2), r^(p-1) == 1 (mod p^2)
p^(r-1) == 1 (mod r^2), r^(q-1) == 1 (mod q^2), q^(p-1) == 1 (mod p^2)
a(9) must have A253683(n) > 121637. - Felix Fröhlich, Jun 18 2016
a(12) must have A253683(n) > 5*10^6. - Giovanni Resta, Jun 20 2016

Wikipedia, Wieferich pair

Cf. A124121, A124122.

Cf. A253683, A253684.

forprime(p=1, , forprime(q=1, p, forprime(r=1, q, if((Mod(p, q^2)^(q-1)==1 && Mod(q, r^2)^(r-1)==1 && Mod(r, p^2)^(p-1)==1) || (Mod(p, r^2)^(r-1)==1 && Mod(r, q^2)^(q-1)==1 && Mod(q, p^2)^(p-1)==1), print1(r, ", ")))))

a(8) from Felix Fröhlich, Jun 18 2016
Name edited by Felix Fröhlich, Jun 18 2016
a(9)-a(11) from Giovanni Resta, Jun 20 2016

cp's OEIS Frontend

A124121 Primes q which are the lesser of a double Wieferich prime pair. (List of known values of q in increasing order, without multiplicity.)

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A244550 a(n) = first odd Wieferich prime to base a(n-1) for n > 1, with a(1) = 2.

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A126432 Double Wieferich primes with q = 5.

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A266829 Primes p such that a prime q < p exists with p^(q-1) == 1 (mod q^2) and q^(p-1) == 1 (mod p^2), i.e., primes that are the larger member of a double Wieferich prime pair.

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A196511 Primes q for which there are primes b < c < p such that b^p == c^p == 1 (mod q^2).

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A196733 Primes q = 2*p+1 for which there are primes b < c < p such that b^p == c^p == 1 (mod q^2).

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A253683 Primes p in increasing order with p > A253684(n) > A253685(n) such that (p, A253684(n), A253685(n)) forms a Wieferich triple.

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A271100 Triangular array read by rows: T(n, k) = k-th largest member of lexicographically earliest Wieferich n-tuple that contains no duplicate members, read by rows, or T(n, k) = 0 if no Wieferich n-tuple exists.

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A253684 Primes q with A253683(n) > q > A253685(n) such that (A253683(n), q, A253685(n)) forms a Wieferich triple.

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