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A349459 Least positive integer m such that the n numbers k^2*(k^2-1) (k=1..n) are pairwise distinct modulo m.

%I A349459 #13 Nov 21 2021 01:16:13
%S A349459 1,5,7,11,13,23,23,23,23,41,41,41,41,41,41,101,101,107,107,107,107,
%T A349459 107,107,107,107,107,107,107,223,223,223,223,223,223,223,223,223,223,
%U A349459 229,239,239,239,383,383,383,383,383,383,383,383,401,401,557,557,557,557,557,557,557,557,557,557,557,733,733,733,733,733,733,733
%N A349459 Least positive integer m such that the n numbers k^2*(k^2-1) (k=1..n) are pairwise distinct modulo m.
%C A349459 Conjecture: For any integer n > 1, the term a(n) is the least prime p > 2*n with p dividing a^2 + b^2 - 1 for no 1 <= a < b <= n.
%C A349459 This has been verified for all n = 2..15000.
%C A349459 Note that a^2*(a^2-1)-b^2*(b^2-1) = (a-b)*(a+b)*(a^2+b^2-1). For any positive integers m and n > 1, if k^2*(k^2-1) (k=1..n) are pairwise distinct modulo m, then it is easy to see that m > 2*n.
%H A349459 Zhi-Wei Sun, <a href="/A349459/b349459.txt">Table of n, a(n) for n = 1..10000</a>
%H A349459 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), no.8, 2794-2812.
%e A349459 a(2) = 5 since the two numbers 1^2*(1^2-1)=0 and 2^2*(2^2-1) = 12 are distinct modulo 5, but they are congruent modulo each of 1,2,3,4.
%t A349459 f[k_]:=f[k]=k^2*(k^2-1);
%t A349459 U[m_,n_]:=U[m,n]=Length[Union[Table[Mod[f[k],m],{k,1,n}]]]
%t A349459 tab={};s=1;Do[m=s;Label[bb];If[U[m,n]==n,s=m;tab=Append[tab,s];Goto[aa]];m=m+1;Goto[bb];Label[aa],{n,1,70}];Print[tab]
%Y A349459 Cf. A000290, A208643.
%K A349459 nonn
%O A349459 1,2
%A A349459 _Zhi-Wei Sun_, Nov 18 2021