This is a classical problem from Number theory. Have you wondered which natural numbers can be expressed as the sum of squares of two integers?
It is easy to notice that all perfect squares are satisfying the above condition, but are they the only ones? The solution to this problem can be justified by the following test:
If a number has a prime factor of the form (4k – 1) to the odd power, then it is impossible to express it as a² + b².
For example, 3,7 & 11 all are primes thus has a factor itself which is of the form (4k-1) where k is 1,2,3 respectively. Thus they cannot be expressed as the sum of squares of two integers.
Infact 3,6,7,11,12,14,15,19,21… Are the first few natural numbers that cannot be expressed as a² + b². But why 6. It is not prime and neither is of the form (4k – 1). But carefully if we notice, it has a prime factor 3 (to the power 1 which is odd) which passes our test. Thus 6 also cannot be expressed.
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