anemone said:Is it possible to write $5^{64}-3^{64}$ as the sum of two squares?
[sp]topsquark said:-DanI'm not familiar with this theorem. Are you saying that any form \(\displaystyle (a^{2p} + b^{2p})(c^{2q} + d^{2q} )(e^{2r} + f^{2r})\) can always be written as the sum of two squares? Or do we additionally need c = e =a, d = f = b or something?
The sum of two squares is a mathematical expression in the form of a² + b², where a and b are integers. This is also known as a Pythagorean triple, representing the sides of a right triangle.
No, not all numbers can be written as the sum of two squares. Numbers that are not themselves perfect squares can only be written as the sum of two squares if they have prime factors that are congruent to 1 modulo 4.
The expression $5^{64}-3^{64}$ is significant because it is a perfect example of a number that can be written as the sum of two squares. Both 5 and 3 are prime numbers congruent to 1 modulo 4, making it possible for them to be written as the sum of two squares.
To determine if a number can be written as the sum of two squares, we need to factor it and check if its prime factors are congruent to 1 modulo 4. If all prime factors meet this condition, then the number can be written as the sum of two squares.
Yes, there is a formula known as the Brahmagupta-Fibonacci identity that can be used to find the two squares that make up the sum of a given number. It states that for any integer n, the sum of two squares can be found by taking the square root of n and then finding the closest integer to it, and then finding the difference between n and the square of that integer.