The fortnightly maths prize question for 26 September was:
5=22+12 and 13=32+22
Make a good guess for a rule about which odd prime numbers can be broken down this way, into a sum of two squares. Make a good guess about which whole numbers generally (not just primes) can be broken down into a sum of two squares.
The 26 September maths prize went unclaimed. Deniz Yukselir had worked out the answer to the first bit (which odd primes are the sum of two square numbers), and Daniel Huang had 90% worked out that answer, but neither wrote up his answer in time.
Answer: The odd primes which equal the sum of two square numbers are the ones which have remainder 1 when divided by 4;
or, equivalently, the ones equal to 4n+1 for some n;
or, equivalently, the ones which have 01 as their last two digits when written in binary rather than decimal.
You can guess this just by checking out the first few primes.
5, 13, 17, 29, 37, 41 (“01-primes” in binary: 4×1+1; 4×3+1; 4×4+1; 4×7+1; 4×90+1; 4×10+1) are equal to the sum of two square numbers
3, 7, 11, 19, 23, 31 (“11-primes” in binary: 4×0+3; 4×1+3; 4×2+3; 4×4+3; 4×5+3; 4×7+3) aren’t.
To guess which numbers in general (not just primes) are equal to the sum of two square numbers, you first need to notice that two sums-of-squares multiplied together make another sum-of-squares.
2 is a sum of squares (2=12+12), so every number whose prime factorisation includes only powers of 2 and powers of 01-primes is a sum of squares.
Multiplying a sum of squares by a square also produces a sum of squares:
So every number whose prime factorisation includes only powers of 2, powers of 01-primes, and even powers of 11-primes, is a sum of squares.
What about the rest? The numbers whose prime factorisations include odd powers of 11-primes?
Try out a few: 6, 14, 15, 21, 22, 33… and you guess that they are not sums of squares.
Proving that the patterns you get from relatively small numbers continue forever is a bit harder: see https://mathsmartinthomas.wordpress.com/2014/08/03/sum-of-two-squares/. But in order even to start proving you must be able to guess the result that you will then prove.
This question was just about the guessing, and you can do the guessing with only year 11 maths.