Re: --Integer solutions for N=a^2+b^2=x^2+y^2+z^2

Gerry wrote :

The sum of two squares N=a^2+b^2

has a solution in three squares N=x^2+y^2+z^2

for every factor q of the sum a+b
for which q-2 is a square.


What is the meaning of such a sentence ???
What is the meaning of
"N has a solution... _for_ every factor of ... " ?

Examples

N=74=5^2+ 7^2=1^2+3^2+8^2

a+b = 12

"for every factor q of a+b" is : for q = { 1, 2, 3, 4, 6 }
then an second "for" is meaningless, or my english is too bad ??
Anyway, q = 4 : q-2 is not a square, neither is q = 6.

Do you mean :
"for every *prime* factor q of a+b, q-2 is a square" ?
(without the meaningless second "for")

or "there exist a factor q of a+b with q-2 being a square" ?
or whatelse ?

A counter example for all these supposed meanings :

N = 325 = 18^2 + 1^2, a+b = 19, prime, and 17 is not a square
N = 17^2 + 6^2, a+b = 23, prime, and 21 is not a square
N = 15^2 + 10^2, a+b = 25 = 5^2, 3 is not a square, 23 is not a square.

But N = 15^2 + 8^2 + 6^2 = 12^2 + 10^2 + 9^2

"N is sum of two squares" and "N is sum of three squares" are two
independant properties.

Regards.

--
Philippe C., mail : chephip+news@xxxxxxx
site : http://chephip.free.fr/ (recreational mathematics)


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