How do I show that there infinitely man integral solutions of an equation?

For example


I used the Euclidean algorithm and then back substituted to find one pair of solution $m=3, n=-22$. But I'm not sure how to find more or prove it.

My textbook hints that - these coefficients are not necessarily produced by euclidean algorithm

  • $\begingroup$ you can just add $264$ to $m$ while subtracting $1947$ from $n.$ Then do it again. And again. $\endgroup$ – Will Jagy Aug 30 '16 at 22:05
  • $\begingroup$ @user1023 Since you found one solution $(m,n) = (3,-22)$, it follows that all (real-valued) solutions are of the form $(m,n) = (3+t, -22 - 1947t/264)$. Can you see a way to choose $t$ so that both $3+t$ and $-22-1947t/264$ are integers? $\endgroup$ – Erick Wong Aug 30 '16 at 22:12
  • $\begingroup$ @ErickWong I'm sorry - I should have mentioned earlier that m,n are integers $\endgroup$ – user1023 Aug 30 '16 at 22:14
  • $\begingroup$ @user1023 My comment was already written with the assumption that $m,n$ are integers. Perhaps you didn't understand it? $\endgroup$ – Erick Wong Aug 30 '16 at 22:15
  • $\begingroup$ @WillJagy thank you - that helps $\endgroup$ – user1023 Aug 30 '16 at 22:17

HINT.-If you have a solution of the equation $ax+by=c$ say $ax_0+by_0=c$ then you have

$$a(x-x_0)+b(y-y_0)=0$$ hence, making $x-x_0=bt$ and $y-y_0=-at$, you have the general solution $$(x,y)=(x_0+bt,\space y_0-at)$$ where $t$ is a parameter.

You have, for instance,$(x,y)=(-13,96)$ (note first that equation reduces to $59x+8y=1$)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.