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Problem: Prove whether or not $x^2+y^2=100,003$ has any integer solutions using modulo arithmetic.

My professor gave the following solution: $x^2+y^2=100,003 \equiv 3 \pmod 4$. By the previous proof verifying that no integer of the form $4k+3$ with integer $k$ is the sum of two squares. Thus $x^2+y^2=100,000$ has no integer solutions.

Why did my professor know to use modulo 4 with remainder 3 looking at the problem? I can't think of any reason to do that.

Lex_i
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    There are a number of useful theorems about squares involving mod-4. For example, an odd prime number is the sum of two squares iff the prime is 3 mod 4. Also, we always start with small mods - it's easier to check all the cases. – Mark Saving Mar 11 '21 at 02:30
  • You may want to add an elementary number theory tag, I guess it is better. – lone student Mar 11 '21 at 03:05

3 Answers3

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  • If $x=2m, y=2n$, then $x^2+y^2$ is always even. Therefore, the equality is not possible.

  • If $x=2m, y=2n-1$ or $x=2m-1, y=2n$, then $$x^2+y^2=4m^2+4n^2-4n+1=4(m^2+n^2-n)+1=100003$$ which is impossible.

There is no need to check $x=2m-1$ and $y=2n$.

  • If $x=2m-1,y=2n-1$ then $x^2+y^2$ is always even. So, it is impossible again.
lone student
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    @Lex_i this is a standard proof method. How many possibilities are there? a positive integer is either even or odd. Yes, I did this because it worked here. If there was a different kind of problem, for example I can take $\left{x,y\right}=3k, 3k-1,3k-2.$ – lone student Mar 11 '21 at 02:55
  • what about the case x, y both being odd? – Lex_i Mar 11 '21 at 03:35
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    @Lex_i Congrats! I forgot that. I'm adding now. Now that you have thought about it, you have completely understood my method.I am glad for that. – lone student Mar 11 '21 at 03:40
  • Please strive not to add more dupe answers to dupes of FAQs – Bill Dubuque Mar 11 '21 at 08:38
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The discriminant of a quadratic form $$A x^2 + Bxy + C y^2$$ is $$ \Delta = B^2-4AC $$ When $\Delta$ is a (positive!) square, the form factors and some things go wrong.

When $\Delta$ is not a square, as your $-4,$ the form represents certain values $\pmod{ |\Delta|}.$ That is, your professor knew to work $\pmod 4$

Here's an example: show that there is no integer solution to $$ x^2 +xy+2y^2 = 70000000003 $$

For forms with positive discriminant (so indefinite), care is needed in treating negative values differently from positive. An example is $x^2-3y^2$ which represents $p$ when prime $p \equiv 1 \pmod {12},$ but instead represents $-p$ when prime $p \equiv 11 \pmod {12}$

Will Jagy
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Why did my professor know to use modulo 4 with remainder 3 looking at the problem? I can't think of any reason to do that.

I'm not sure I understand your question properly. You mentioned that:

By the previous proof verifying that no integer of the form 4k+3 with integer k is the sum of two squares

That was already proven, so you simply need to verify if the two squares are in the form, so by taking mod 4, you'll be left with the remainder. ie $4k + r \equiv r$ (mod 4) so you want $r = 3$ if you want to prove no solution exists.

spectralblue
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