0

Is it possible to prove or disprove by modular arithmetic that $$f=\frac{(19(8m\pm1)+1)^2+(22(8m\pm1)+1)^2}{9}$$ can never be a perfect square ($m$ is any integer)? More generally, can a condition be derived on some $$f=\frac{(t(8m\pm1)+1)^2+(s(8m\pm1)+1)^2}{(s-t)^2}$$ to be an odd perfect square? ($t$ is always odd)

RTn
  • 299
  • If this is what you're trying to prove, I think you can ignore the denominator, provided the quotient is an integer. – Matt Samuel May 28 '19 at 19:13
  • Yes, sure.. But isn't the denominator is necessary to prove it cannot be an integer? – RTn May 28 '19 at 19:22
  • If you want to prove it isn't an integer, show the numerator is not 0 mod 9. – Matt Samuel May 28 '19 at 19:24
  • No, I probably did not word it correctly. I want to prove that it isn't a perfect square. But if that is the case, it is works only if the function is 0 mod 9, right? So, I dont understand why you suggest to discard the denominator? Could you please explain? – RTn May 28 '19 at 19:28
  • 1
    If $n$ is an integer and $s\neq t$, then $n$ is a perfect square if and only if $(s-t)^2n$ is a perfect square. So if you know the quotient is an integer, then it is a perfect square if and only if the numerator is. This is actually also true even if the quotient isn't an integer, as it is meaningful for a rational number to not be a perfect square. – Matt Samuel May 28 '19 at 19:31
  • Ah I understand now. Thanks a lot. I am very new to modular arithmetic, so could you please help out in proving the example I have cited above? That way, I can have a base to refer to for future cases. – RTn May 28 '19 at 19:37
  • Which of the following do you not know: linear polynomials, equivalence relations, grade school arithmetic rules ? I might be willing to help. –  May 29 '19 at 00:55
  • @Roddy MacPhee , Ah, that is so generous of you! I did the following.. For $f=\left(\frac{19(8m-1)+1}{3}\right)^2+\left(\frac{22(8m-1)+1}{3}\right)^2$ gives $f=\left(\frac{152m-18}{3}\right)^2+\left(\frac{176m-21}{3}\right)^2$ which reduces to $f=(2m_1-6)^2+(2m_2-7)^2$ for some $m_1$, $m_2$. After this I am stuck as to how to prove that $f$ can never be a perfect square. – RTn May 29 '19 at 07:06
  • Lets [https://chat.stackexchange.com/rooms/94237/rtn][chat] –  May 29 '19 at 10:04
  • Sure... Can you approve me in the chat room to type please? – RTn May 29 '19 at 10:19

0 Answers0