A349530 Least positive integer m such that the n numbers k*(k^4+1) (k=1..n) are pairwise distinct modulo m^2.

1, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 25, 25, 25, 25, 25

Offset: 1

Author

Zhi-Wei Sun, Nov 21 2021

Keywords

• nonn

Comments

Conjecture 1: Suppose that 5^a < sqrt(n) <= 5^(a+1). Then a(n) = 3*5^a if sqrt(n) <= sqrt(3)*5^a, and a(n) = 5^(a+1) otherwise.

Conjecture 2: Let f(n) be the least positive integer m such that the n numbers 18k*(k^4+1) (k=1..n) are pairwise distinct modulo m^2. Then f(n) is the least power of 5 not smaller than sqrt(n), except for 25 < n <= 45 (and in this case f(n) = 19).

Conjecture 3: Let n be a positive integer not among 26, 27, 28, 626, 627, 628, 629, 630, and define D(n) as the least positive integer m such that the n numbers k*(k^4+1) (k=1..n) are pairwise distinct modulo m. If 5^a < n <= 3*5^a, then D(n) = 3*5^a. If 3*5^a < n <= 5^(a+1), then D(n) = 5^(a+1).

We have verified the above conjectures for n up to 10^5.

Examples

			a(2) = 3 since the two numbers 1*(1^4+1) = 2 and 2*(2^4+1) = 34 are distinct modulo 3^2, but they are congruent modulo each of 1^2 and 2^2.
		

Links

• Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.
• Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.
• Quan-Hui Yang and Lilu Zhao, On a conjecture of Sun involving powers of three, arXiv:2111.02746 [math.NT], 2021.

Crossrefs

Cf. A000351, A002523, A208643, A349459, A349537.

Programs

Mathematica

f[k_]:=f[k]=k*(k^4+1);
U[m_,n_]:=U[m,n]=Length[Union[Table[Mod[f[k],m^2],{k,1,n}]]]
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,1,80}];Print[tab]