login
A316976
Numbers k such that some of the values (r0-r1+k) mod k for all pairs (r0,r1) of quadratic residues mod k are unique.
0
1, 3, 4, 5, 8, 9, 12, 15, 16, 20, 24, 32, 36, 40, 45, 48, 60, 64, 72, 80, 96, 120, 128, 144, 160, 180, 192, 240, 288, 320, 360, 384, 480, 576, 640, 720, 960, 1152, 1440, 1920, 2880, 5760
OFFSET
1,2
COMMENTS
These are the numbers k such that A316975(k) = 1.
It is conjectured that this list is finite and limited to the terms given in the DATA section.
All known terms are 5-smooth.
EXAMPLE
The quadratic residues mod 12 are 0, 1, 4 and 9. For each pair (r0,r1) of these quadratic residues, we compute (r0-r1+12) mod 12, leading to:
0 1 4 9
+------------
0 | 0 11 8 3
1 | 1 0 9 4
4 | 4 3 0 7
9 | 9 8 5 0
The values 1, 5, 7 and 11 are unique in the above table. Therefore 12 belongs to the list.
MATHEMATICA
Select[Range[10^3], Function[{n}, Min@ Tally[#][[All, -1]] == 1 &@ Flatten[Mod[#, n] & /@ Outer[Subtract, #, #]] &@ Union@ PowerMod[Range@ n, 2, n]]] (* Michael De Vlieger, Jul 20 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Arnauld Chevallier, Jul 17 2018
STATUS
approved