mathschallenge.net

Problem

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?

Solution

By working through the primes the following list can be obtained:

2 = 1 + 1	29 = 4 + 25	61 = 25 + 36
5 = 1 + 4	37 = 1 + 36	73 = 9 + 64
13 = 4 + 9	41 = 16 + 25	89 = 25 + 64
17 = 1 + 16	53 = 4 + 49	97 = 16 + 81

That is, there are twelve Pythagorean primes under 100.

Make a complete list of all the primes less than 100 and circle the Pythagorean primes. With the exception of 2, what do you notice?

How many different ways can you write each of the Pythagorean primes as the sum of two square?