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

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Squares Sum
AI Thread Summary
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.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Is it possible to write $5^{64}-3^{64}$ as the sum of two squares?
 
Mathematics news on Phys.org
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!
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Just chatting with my son about Maths and he casually mentioned that 0 would be the midpoint of the number line from -inf to +inf. I wondered whether it wouldn’t be more accurate to say there is no single midpoint. Couldn’t you make an argument that any real number is exactly halfway between -inf and +inf?
Back
Top