(Number theory) Sum of three squares solution proof

I think the phrase "without loss of generality" refers to the fact that you can always find a solution by permuting the numbers x, y, z, so you don't need to care about any other order than the one I just mentioned.The rest of the proof is a bit terse, but I think it is basically correct. There are just a few small things that could be improved:- You call it an "informal proof attempt". If you want to make it more formal, you could start by stating what you are going to prove. In this case, it is something like "The only integer solutions to x^2 + y^2 + z^2 = 51 are the permutations of ##\pm1
  • #1
Xizel
4
0

Homework Statement



Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality."

Homework Equations



The Attempt at a Solution



My informal proof attempt:

Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4
Then x2, y2, y2 = (0 or 1) mod 4
So x2 + y2 + z2 = [ (0 or 1) + (0 or 1) + (0 or 1) ] mod 4
Since 51 = 3 (mod 4) = x2 + y2 + z2, then x, y, z = (1 or 3) mod 4

It is obvious that the solution is the permutations of ##\pm1, \pm1, \pm7##.

It's my first proof course and I'm a little shaky. Is my logic correct? I feel like I took a leap from "Since 51..." to the solution, is there a more formal way to write that? I'm also not sure how to use wlog here. Thanks.
 
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  • #2
Xizel said:
It is obvious that the solution is the permutations of ##\pm1, \pm1, \pm7##.
I have a additional solution.
 
  • #3
fresh_42 said:
I have a additional solution.

If my logic is correct, then x, y, z can take on values of 1, 3, 5, 7. I gave it some thought and I'm not seeing it :nb)
 
  • #4
##2\cdot 25 + 1##
The remark "use w.l.o.g." probably refers to the assumption ##x \geq y \geq z \geq 0## which you could make.
 

Related to (Number theory) Sum of three squares solution proof

1. How is the sum of three squares solution proved in number theory?

In number theory, the sum of three squares solution is proved through a mathematical technique called Fermat's infinite descent. This method involves assuming the existence of a solution and then proving that it leads to a contradiction, thus showing that the original assumption was false. This process is repeated infinitely until all possible solutions have been exhausted.

2. What is the significance of the sum of three squares solution in number theory?

The sum of three squares solution is significant in number theory because it provides a way to represent any positive integer as the sum of three squares. This has important implications in various areas of mathematics, including the study of prime numbers and Diophantine equations.

3. Can the sum of three squares solution be extended to higher powers?

No, the sum of three squares solution cannot be extended to higher powers. In fact, it is one of only two cases (along with the sum of two squares) where a similar solution exists. For powers higher than three, no such solution has been found and it is believed that none exists.

4. How does the sum of three squares solution relate to Pythagorean triples?

The sum of three squares solution is closely related to Pythagorean triples, which are sets of three positive integers that satisfy the Pythagorean theorem (a^2 + b^2 = c^2). This is because any Pythagorean triple can be expressed as the sum of three squares, and conversely, any sum of three squares can be written as a Pythagorean triple.

5. Are there any practical applications of the sum of three squares solution?

The sum of three squares solution has several practical applications, particularly in the field of cryptography. It is used in algorithms for generating random numbers and in error-correcting codes. It also has connections to other areas such as coding theory and signal processing.

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