MHB Sum of Two Squares: Can $5^{64}-3^{64}$ Be Written?

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The discussion centers on whether the expression $5^{64}-3^{64}$ can be represented as the sum of two squares. Participants reference the identity that states the product of two sums of squares can also be expressed as a sum of two squares. Dan expresses gratitude for contributions from others, including Kali and Opalg, who provided explanations and solutions. The conversation highlights the collaborative effort to explore this mathematical question. Ultimately, the focus remains on the mathematical properties related to sums of squares.
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Is it possible to write $5^{64}-3^{64}$ as the sum of two squares?
 
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anemone said:
Is it possible to write $5^{64}-3^{64}$ as the sum of two squares?

$5^{64}-3^{64}$
=$(5^{32}+3^{32})(5^{16}+3^{16})(5^8+3^8)(5^4+3^4)(5^2+3^2)(5+3)(5-3)$
= $16((5^{32}+3^{32})(5^{16}+3^{16})(5^8+3^8)(5^4+3^4)(5^2+3^2)$
= $4^2((5^{32}+3^{32})(5^{16}+3^{16})(5^8+3^8)(5^4+3^4)(5^2+3^2)$

as product of 2 numbers both sum of 2 squares can be represented as sum of 2 squares and 4^2 is a square so argument repeatedly we have the ans Yes
 
I'm not familiar with this theorem. Are you saying that any form [math](a^{2p} + b^{2p})(c^{2q} + d^{2q} )(e^{2r} + f^{2r})[/math] can always be written as the sum of two squares? Or do we additionally need c = e =a, d = f = b or something?
-Dan
 
topsquark said:
I'm not familiar with this theorem. Are you saying that any form [math](a^{2p} + b^{2p})(c^{2q} + d^{2q} )(e^{2r} + f^{2r})[/math] can always be written as the sum of two squares? Or do we additionally need c = e =a, d = f = b or something?
-Dan
[sp]
The identity $(a^2+b^2)(c^2+d^2) = (ac+bd)^2 + (ad-bc)^2$ shows that a product of sums of two squares is also a sum of two squares.
[/sp]
 
(Doh) I knew that one.

Thanks!

-Dan
 
Thanks Kali for your solution!

And thanks to Opalg too for explaining thing for topsquark! I appreciate that!
 
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