Fermat's theorem on sums of two squares composite number Suppose that there is some natural number $a$ and $b$. Now we perform $c = a^2 + b^2$. This time, c is even. 
Will this $c$ only have one possible pair of $a$ and $b$?
edit: what happens if c is odd number?

Not necessarily. For example, note that $50=1^2+7^2=5^2+5^2$, and $130=3^2+11^2=7^2+9^2$. For an even number with more than two representations, try $650$. 
We can produce odd numbers with several representations as a sum of two squares by taking a product of several primes of the form $4k+1$.  To get even numbers with multiple representations, take an odd number that has multiple representations, and multiply by a power of $2$. 
To help you produce your own examples, the following identity, often called the Brahmagupta Identity, is quite useful:
$$(a^2+b^2)(x^2+y^2)=(ax\pm by)^2 +(ay\mp bx)^2.$$ 
