Small Solutions of the Legendre Equation

✍ Scribed by Todd Cochrane; Patrick Mitchell

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 157 KB
• Volume: 70
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2221

✦ Synopsis

Holzer, using a deep theorem, proved that if the Legendre equation ax 2 +by 2 & cz 2 =0 is in normal form and has a nonzero solution then there is a solution with |x| -bc, |y| -ac and |z| -ab. The first elementary proof of this result was given by Mordell with a small gap filled by Williams. We give a new elementary proof of this theorem. We show in fact that the lattice of solutions of the congruence ax 2 +by 2 &cz 2 #0 (mod abc) that is constructed in most textbook proofs of Legendre's theorem always contains such a solution.