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• ... False. If you allow zero to be a square, then 3^2 + 0^2 is a counterexample. If you do not allow zero to be a square then 3^2 + 6^2 is a counterexample.
Message 1 of 3 , Dec 4 12:07 PM
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Kermit Rose <kermit@...> wrote:

> If the sum of two integer squares is composite,
> it can be factored into prime integers, each of which
> is the sum of two squares.

False.

If you allow zero to be a square, then
3^2 + 0^2 is a counterexample.
If you do not allow zero to be a square then
3^2 + 6^2 is a counterexample.
Either way you claim is clearly wrong.

David
• ... If d = 1 mod 4, then the integers of the field k(sqrt(d)) are indeed of your form but you must also allow A and B to be both half-integers. See for example
Message 2 of 3 , Dec 4 11:19 PM
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--- In primenumbers@yahoogroups.com, Kermit Rose <kermit@...> wrote:
>
>
> Primes in algebraic number systems:
>
> Consider the extension to the integers,
>
> {A + B * sqrt(d) }, where A, B are variable integers, and d is a
> fixed integer.

If d = 1 mod 4, then the integers of the field k(sqrt(d)) are indeed of your form but you must also allow A and B to be both half-integers.
See for example Hardy & Wright, Theorem 238.

Mike
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