A350540 a(n) = smallest number x such that x^2 == 17 (mod 2^n).

0, 1, 1, 1, 1, 7, 9, 23, 23, 23, 233, 279, 279, 1769, 1769, 6423, 9961, 9961, 55575, 55575, 206569, 206569, 842007, 1255145, 2939159, 2939159, 2939159, 2939159, 64169705, 64169705, 204265751, 204265751, 869476073, 869476073, 3425491223, 3425491223, 13754377961

Offset: 0

Author

Tejo Vrush, Jan 04 2022

Keywords

• nonn

Comments

17 is the smallest nonsquare that is congruent to a square mod 2^n for any n.

Any number that is congruent to a square mod 2^n for any n is of the form (4^a)*(8b+1). Such numbers have density 1/6.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000290, A000079.

Programs

Mathematica

Table[PowerMod[17,1/2,2^k],{k,0,36}] (* Giorgos Kalogeropoulos, Jan 31 2023 *)

PARI

a(n) = my(x=0); while (Mod(x, 2^n)^2 != 17, x++); x; \\ Michel Marcus, Jan 04 2022

Python

from sympy.ntheory import sqrt_mod
def A350540(n): return min(sqrt_mod(17,2**n,all_roots=True)) # Chai Wah Wu, Jan 12 2022

Extensions

a(13)-a(28) from Michel Marcus, Jan 04 2022

a(30)-a(36) from Alois P. Heinz, Jan 04 2022

Edited by N. J. A. Sloane, Jan 12 2022