Theorem 6 10 Jacobsthal Assume That Q 1 Mod 4 For A F Consider The Sum S A O X Ax Xef Then The Follow 1 (57.24 KiB) Viewed 53 times
Theorem 6.10 (Jacobsthal). Assume that q = 1 mod 4. For a € F*, consider the sum S(a) = [o(x³ + ax). xEF Then the following hold: (i) S(a) is an even integer, whose absolute value only depends on whether a is a square or not; (ii) if 2A and 2B denote the values of S() on squares, respectively on non- squares, then the positive integers A and B satisfy A²+ B² = q.
Theorem 6.11. The Paley graph P(q) is strongly regular: any two adjacent vertices have a =(q - 5) common neighbours, and any two non-adjacent vertices have c = (q-1) common neighbours.
Proposition 6.13. The Paley graph P(q²) has chromatic number q, and indepen- dence number q.
Theorem 6.14. The bi-Paley graph BP(q) is a design graph, with parameters m=q, d = (q-1), c=(q-3).
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