Prime Square Sums
Number Theory
Level
4
The prime number 13 is called a Pythagorean prime because it can be written as the sum of two square numbers.
2² + 3² = 4 + 9 = 13.
How many Pythagorean primes are there below 100?
The answer is 12.
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1 solution
Consider the odd primes first. The only primes that work are the primes that are 1 mod 4. There are 11 of these that are less than 100, namely 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, and 97.
Consider the even primes, of which there is only one, namely 2. Clearly 2=1+1, hence it is a pythagorean prime.
Thus, there are 12 of them.