## Re: sum of the squares of primes equalling a square

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• ... Phil wisely chose to use 24 = 2^3 * 3 for a modular exhaustion. When I first encountered number theory, it took me a while to understand why working mod 8
Message 1 of 7 , Jul 6, 2012
Phil Carmody <thefatphil@...> wrote:

> Most likely attack is mod 24.
....

Phil wisely chose to use 24 = 2^3 * 3
for a modular exhaustion.

When I first encountered number theory,
it took me a while to understand why
working mod 8 was important, when it seemed
to me (wrongly) that we need only care

If one will glance, for example, at
http://en.wikipedia.org/wiki/Kronecker_symbol
then one will see that mod 8 criteria really matter.
[Also when trying to extract modular square roots.]

Silly summary: when dealing with any even prime,
it may be wise to work modulo its cube.

David
• ... Thank you Phil, that was very helpful. Now if it could only level a house made from hypothetical perfect euler bricks! (That happens to be the problem
Message 2 of 7 , Jul 7, 2012
> Whenever there's a puzzle involving squares, 24 is a very powerful lever.

Thank you Phil, that was very helpful. Now if it could only level a house made from hypothetical perfect euler bricks! (That happens to be the problem that led me down this particular sum of squares of primes theme, albeit as a side diversion!)

Mark

--- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
>
> --- 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}
>
>
> > 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
>
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