13 | May | 2008 | Soliloquy

Archive for May 13th, 2008

All square numbers have odd number of factors. All other numbers have even number of factors.
To check whether a number is prime, it is sufficient to check its divisibiliy by all prime numbers less than square root of the number.
A number n is a sum of two squares if and only if all prime factors of n of the form 4m+3 have even exponent in the prime factorization of n.
if p is a prime number, then for any integer a, $a p - a$ will be evenly divisible by p. This can be expressed in the notation of modular arithmetic as follows.
a^p ≡ a (mod p) . This is Fermat’s Little Theorem.
If an integer $n$ is greater than 2, then the equation $a n + b n = c n$ has no solutions in non-zero integers $a$ , $b$ , and $c$ . This is Fermat’s Last Theorem.