A German mathematician Christian Goldbach (1690-1764) stated that every odd composite number can be written as a sum of a prime and twice a square.

Some examples:
9 = 7 + 2*1²
21 = 3 + 2*3²
25 = 7 + 2*3²
33 = 31 +2*1²

Now we know that his conjecture is false.

What is the smallest odd composite number not complying with C.G.'s conjecture?

Unfortunately I don't think there are any methods not using brute force to solve this.  But searching for the numbers found by Charlie gives

http://oeis.org/A060003

Posted by Jer on 2013-02-06 12:33:29