ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 Clarke, Francis, University of Wales Swansea, UK Closed Formulas for representing primes by binary quadratic forms It is a result due to Gauss that if p = 4k + 1 is prime and p = x^{2} + y^{2}, where x and y are integers and x is odd, with the sign determined by x ≡ 1 mod 4, then

x ≡ \frac{1}{2}\binom{2k}{k} mod p.

We discuss various generalisations. For example if p is prime and 4p = x^{2} + 19y^{2}, where the sign of x is chosen so that x is a quadratic residue modulo 19, then

x ≡ (−1)^{(p+1)/2} ∑_{4a+6b=p−1} \binom{2a + 2b}{a + b}\binom{a + b}{a}2^{3a+2b}19^{a+2b} mod p.