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A308376
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Least positive integer k < prime(n) such that k + k' is a square, where k' is the unique integer with 0 < k < prime(n) and k*k' == 1 (mod prime(n)), or 0 if such a number k does not exist.
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1
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0, 2, 0, 0, 0, 2, 3, 3, 17, 16, 7, 10, 12, 0, 4, 28, 6, 19, 2, 22, 5, 0, 4, 12, 11, 7, 18, 19, 8, 6, 4, 45, 3, 11, 8, 58, 2, 11, 43, 37, 4, 17, 28, 5, 27, 7, 3, 8, 10, 69, 3, 4, 14, 38, 6, 30, 12, 48, 49, 27, 2, 7, 4, 22, 24, 31, 13, 60, 5, 16, 3, 9, 15, 23, 3, 4, 10, 16, 14, 9, 91, 53, 87, 39, 62, 12, 6, 30, 49, 26, 4, 7, 25, 26, 14, 13, 15, 35, 16, 18
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OFFSET
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1,2
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COMMENTS
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Conjecture 1: a(n) = 0 only for n = 1, 3, 4, 5, 14, 22. In other words, if p is a prime not among 2, 5, 7, 11, 43, 79, then there are k and m among 1,...,p-1 with k*m == 1 (mod p) such that k + m is a square.
We have verified this for all primes p < 4*10^7.
Conjecture 2: For any prime p > 18, there are k and m among 1,...,p-1 with k*m == 1 (mod p) such that k + m is a triangular number.
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LINKS
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EXAMPLE
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a(9) = 17 with 17*19 == 1 (mod prime(9)=23) and 17 + 19 = 6^2.
a(16) = 28 with 28*36 == 1 (mod prime(16)=53) and 28 + 36 = 8^2.
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MATHEMATICA
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p[n_]:=p[n]=Prime[n]; SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[Do[If[SQ[k+PowerMod[k, -1, p[n]]], tab=Append[tab, k]; Goto[aa]], {k, 1, p[n]-1}]; tab=Append[tab, 0]; Label[aa], {n, 1, 100}]; Print[tab]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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