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URL: https://oeis.org/A353077

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A353077
Triangle read by rows, where the n-th row consists of the lexicographically earliest solution for n integers in 0..p-1 whose n*(n-1) differences are congruent to 1..p-1 (mod p), where p=n*(n-1)+1. If no solution exists, the n-th row consists of n -1's.
3
0, 0, 1, 0, 1, 3, 0, 1, 3, 9, 0, 1, 4, 14, 16, 0, 1, 3, 8, 12, 18, -1, -1, -1, -1, -1, -1, -1, 0, 1, 3, 13, 32, 36, 43, 52, 0, 1, 3, 7, 15, 31, 36, 54, 63, 0, 1, 3, 9, 27, 49, 56, 61, 77, 81, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 1, 3, 12, 20, 34, 38, 81, 88, 94, 104, 109
OFFSET
1,6
COMMENTS
Comment from Martin Becker, May 18 2025: (Start)
Rows k with k-1 not a prime power are precisely the rows with -1 values for k <= 2*10^10. Cf. the Gordon (2020) link.
In the b-file, values a(n) > 3 from row 17 onwards depend on the conjecture that all perfect difference sets are Singer type, and were obtained through computer enumeration of Singer type sets.
(End)
LINKS
Leonard E. Dickson, Problem 142, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), pp. 107-108.
Daniel Gordon, On difference sets with small lambda, arXiv:2007.07292 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Perfect Difference Set
EXAMPLE
n row
1 [0];
2 [0,1];
3 [0,1,3];
4 [0,1,3,9];
5 [0,1,4,14,16];
6 [0,1,3,8,12,18];
7 no solution exists;
8 [0,1,3,13,32,36,43,52];
9 [0,1,3,7,15,31,36,54,63];
10 [0,1,3,9,27,49,56,61,77,81];
11 no solution exists;
12 [0,1,3,12,20,34,38,81,88,94,104,109];
13 no solution exists;
14 [0,1,3,16,23,28,42,76,82,86,119,137,154,175];
15 no solution exists;
16 no solution exists.
PROG
(PARI) isok(n, v) = my(p=n*(n-1)+1); setbinop((x, y)->lift(Mod(x-y, p)), v, v) == [0..p-1];
row(n) = forsubset([n^2-n+1, n], s, my(ds = apply(x->x-1, Vec(s))); if (isok(n, ds), return(ds)); );
CROSSREFS
KEYWORD
sign,look,tabl
AUTHOR
Michel Marcus, Apr 22 2022
EXTENSIONS
Name and data corrected for "lexicographically earliest solution" by Michel Marcus, May 09 2022
Adjusted to a regular triangle, and rows 1, 2, 7, and 10-12 inserted by Pontus von Brömssen, May 09 2022
STATUS
approved