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A225577
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Least integer m>1 such that 1^2,2^2,...,n^2 are pairwise incongruent modulo 2^m-1.
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1
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2, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 13, 13, 13, 13, 13, 13, 13
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OFFSET
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1,1
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COMMENTS
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Conjecture: a(n) is the least prime p such that 2^p-1 is a Mersenne prime greater than 2n-1.
This conjecture implies that there are infinitely many Mersenne primes.
Zhi-Wei Sun also conjectured that for each n>17 the least Fibonacci number modulo which 1^2,2^2,...,n^2 are pairwise incongruent is just the first Fibonacci prime greater than 2n-1.
This phenomenon might happen for some other Lucas sequences u_0,u_1,... given by u_0 = 0, u_1 = 1, and u_{k+1} = A*u_k-B*u_{k-1} for k>0, with A>0 and B (nonzero) relatively prime and A^2 > 4B.
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LINKS
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EXAMPLE
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a(4)=5 since 1^2,2^2,3^2,4^2 are incongruent modulo 2^5-1=31, but 1^2==4^2 (mod 2^4-1), 3^2==4^2 (mod 2^3-1) and 2^2==4^2 (mod 2^2-1).
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MATHEMATICA
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R[n_, m_]:=Union[Table[Mod[k^2, m], {k, 1, n}]]
s=2
Do[Do[If[Length[R[n, 2^m-1]]==n, s=m; Print[n, " ", m]; Goto[aa]], {m, s, 100000}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 20}]
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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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