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.

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

Author

Felix Fröhlich, Mar 30 2016

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
  ....
		

Links

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

Crossrefs

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

Programs

PARI

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.

PARI

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

Sage

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

Extensions

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

More terms from Bruce Leenstra, May 18 2016