- True. And those identities from William were pretty neat.

I had said,

"If the sum of the squares of 5 primes equals a square, two of the primes are 2

and 3"

which was a little vague. It should really have been

If the sum of the squares of 5 primes equals a square, one of the primes is 2 and another is 3.

Besides the previous

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

2.

we now have this:

If the sum of the squares of 11 primes equals a square, two of the primes are 2 and another is 3.

I never would have guessed off the top of my head that with this many squares there would be such restrictions!

Mark

--- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:

>

> Yes, the sum of squares of 3 composites can equal a square.

>

> But Mark stated that the sum of squares of 3 primes cannot;

> it's pretty easy to prove. (Choose the right modulus for

> solving your Diophantine equation and it becomes clear.)

>

> On 7/6/2012 8:53 AM, elevensmooth wrote:

> >

> >

> > --- In primenumbers@yahoogroups.com, "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.

> >

> > 8^2 + 49^2 + 64^2 = 81^2

> >

> > For any integers p, q, s, and t let

> >

> > d=(p^2+q^2+s^2+t^2)/2

> > c=(p^2+q^2-s^2-t^2)/2

> > a=ps+qt

> > b=pt-qs

> >

> > then a^2 + b^2 + c^2 = d^2

> >

> > Depending on the choices, c and d might be halves, and all the values will need be doubled to get integers.

> >

> > William Lipp

> > Poohbah of OddPerfect.org

> >

> >

> >

> >

> > ------------------------------------

> >

> > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com

> > The Prime Pages : http://primes.utm.edu/

> >

> > Yahoo! Groups Links

> >

> >

> >

> >

> >

> - --- On Fri, 7/6/12, Mark <mark.underwood@...> wrote:
> Some tidbits that some may find interesting and perhaps fun to prove:

Most likely attack is mod 24.

>

> 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)

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

Presuming the above attack works too:

> of the primes are 2.

\ 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 - --- In primenumbers@yahoogroups.com,

Phil Carmody <thefatphil@...> wrote:

> Most likely attack is mod 24.

....

> Your result is confirmed.

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

about mod 2.

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 > 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}

>

> 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

>