Posted by: Gary Ernest Davis on: June 11, 2011

Completing the square A former colleague wondered if the quadratic formula held when are complex numbers. He could, of course, have figured this out by the process of completing the square: If then exactly when (thanks to James Tanton for this trick). This happens exactly when or, equivalently when . This leads us to the […]

Posted by: Gary Ernest Davis on: May 16, 2011

Catalan pseudoprimes As a special case of Fermat’s little theorem, if is an odd prime number then is divisible by , or, in the language of congruences, . The converse of this is not true: if where is a positive integer, it does not follow that is prime. For example could be . There are […]