Writing a number as sum of two triangular numbers?

Fermat had proved that any number can be written as sum of three triangular numbers.





Question: What type of numbers ';can not'; be written as sum of TWO triangular numbers.Writing a number as sum of two triangular numbers?
wpw, i have thumbs you up, but i cannot find where is the proof of this question..





however:


Lemma- the equation n=a^2+b^2 has solution in integers iff there are a even number of p=3(mod4) that divides n.


(the proof can be find in any elementary number theory book, however it is only a corollary that -1 is a quadratic residue mod p iff 4|p-1).


Now we must have the equation (*) n= a(a+1)/2 + b(b+1)/2.


Let A=2a+1, and B=2b+1: we have (*) iff 8n+2=A^2+B^2.


now suppose that a prime p=3 mod 4 divides 8n+2, then it divides also A^2+B^2, then -1 is a quadratic residue mod p, contradiction.


So, every (and only) numbers n such that 4n+1 has a even number of prime p=3mod4 that divides itself can be written as sum of two triangular numbers.Writing a number as sum of two triangular numbers?
* 2939. On the Representation of Numbers as Sums of Triangular Numbers


* U. V. Satyanarayana


* The Mathematical Gazette, Vol. 45, No. 351 (Feb., 1961), pp. 40-43


(article consists of 4 pages)


* Published by: The Mathematical Association


* Stable URL: http://www.jstor.org/stable/3614771
what r triangular numbers?