A225577 Least integer m>1 such that 1^2,2^2,...,n^2 are pairwise incongruent modulo 2^m-1.

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

Offset

1,1

Comments

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.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Zhi-Wei Sun, The least modulus for which consecutive polynomial values are distinct, arXiv:1304.5988 [math.NT], 2013-2015.

Zhi-Wei Sun, On functions taking only prime values, J. Number Theory, 133 (2013), 2794-2812.

Example

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).

Mathematica

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}]

Crossrefs

Cf. A000043, A000668, A001605, A005478.

Sequence in context: A347293 A113730 A154404 * A265531 A083662 A256405

Adjacent sequences: A225574 A225575 A225576 * A225578 A225579 A225580

Keyword

nonn

Author

Zhi-Wei Sun, May 10 2013

Status

approved