From: cet1-nospam@cam.ac.uk.invalid (Chris Thompson) Subject: Re: number = sum of 3 triangular #'s Date: 21 Jun 2000 20:42:41 GMT Newsgroups: sci.math Summary: [missing] In article , Jim Ferry wrote: >ceaton1@my-deja.com wrote: >> >> I think I recently saw a reference here to a proof that any positive >> integer could be expressed as the sum of three triangular numbers. I >> would be grateful if someone could repeat that reference (if it exists). > >I don't know the reference or proof, but there is some information about >the general case (any number = sum of 5 pentagonal numbers, etc.) at > >http://mathworld.wolfram.com/FermatsPolygonalNumberTheorem.html > >The general theorem is called "Fermat's Polyhonal Number Theorem". It was >proved by Cauchy in 1813. (The triangular case was proved by Gauss.) I think it is more commonly called "Cauchy's Polygonal Number Theorem", just as the four-squares case is called Lagrange's Theorem. Fermat certainly *claimed* it as a theorem, but this is familiar territory: | The theorem is based on the most diverse and abstruse mysteries of | numbers, but I am not able to include the proof here... as [1] translates him! The triangular numbers case is easily seen to be equivalent to the 8k+3 case of the theorem, due to Gauss, that a positive integer N is the sum of three squares if and only if it is not of the form 4^a (8k+7). This is usually mentioned, but rarely proved, in introductory textbooks on Number Theory. Nathanson includes a proof (following Landau's) in the first chapter of [1], where he also covers Cauchy's result. He uses but does not prove Dirichlet's theorem on primes in arithmetic progressions, so you might also want something that provides a proof of that, e.g. [2]. [1] Melvyn B. Nathanson "Additive Number Theory - The Classical Bases" (Springer, 1996) [GTM 164] [2] Jean-Pierre Serre "A Course In Arithmetic" (Springer, 1973) [GTM 7] Chris Thompson Email: cet1 [at] cam.ac.uk