--- On Fri, 7/6/12, Mark <

mark.underwood@...> wrote:

> Some tidbits that some may find interesting and perhaps fun to prove:

>

> The sum of the squares of 3 primes never equals a square.

>

> If the sum of the squares of 5 primes equals a square, two

> of the primes are 2 and 3.

>

> (For example 2^2 + 3^2 + 5^2 + 7^2 + 13^2 = 16^2)

Most likely attack is mod 24.

Squares are {0,1,4,9,16,12}.

Prime squares are {1} and the trivial (choice-free) {4,9}

So we have some 2s, some 3s, and then some things that square to 1:

\ 0*2 1*2 2*2 3*2 4*2 5*2

0*3 0+5 4+4 8+3 12+2 16+1 20+0 | no solns

1*3 9+4 13+3 17+2 21+1 1+0 x | 13+3 and 1+0 are squares

2*3 18+3 22+2 2+1 6+0 x x | 22+2 is a square

3*3 3+2 7+1 11+0 x x x | no solns

4*3 12+1 16+0 x x x x | 16+0 is a square

5*3 21+0 x x x x x | no solns

13+3 is {2, 3, p, q, r}

1+0 is {2, 2, 2, 2, 3}

22+2 is {2, 3, 3, p, q}

16+0 is {2, 3, 3, 3, 3}

Your result is confirmed.

> If the sum of the squares of 7 primes equals a square, three

> of the primes are 2.

Presuming the above attack works too:

\ 0*2 1*2 2*2 3*2 4*2 5*2 6*2 7*2

0*3 0+7 4+6 8+5 12+4 16+3 20+2 0+2 4+2 | 12+4 has 3*2

1*3 9+6 13+5 17+4 21+3 1+2 5+1 9+0 x | 21+3 and 9+0 have >=3*2

2*3 18+5 22+4 2+3 6+2 10+1 14+0 x x | no solns

3*3 3+4 7+3 11+2 15+1 19+0 x x x | 15+1 has 3*2

4*3 12+3 16+2 20+1 0+0 x x x x | 0+0 has 3*2

5*3 21+2 1+1 5+0 x x x x x | no solns

6*3 6+1 10+0 x x x x x x | no solns

7*3 15+0 x x x x x x x | no solns

Confirmed.

Whenever there's a puzzle involving squares, 24 is a very powerful lever.

Phil