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A271100
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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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4
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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;
....
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PROG
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\\finds candidate values for the highest value of a tuple up to some value u.
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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