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Find the prime numbers $p<q<r$ such that $r^2-q^2-p^2$ is a perfect square.

I think the only solution is (2,3,7) but i cannot prove it. The equation would be $r^2-q^2-p^2=k^2$ equivalently $q^2+p^2+k^2=r^2$ which is somehow a classical diophantine equation (of the pythagorean quadruples) which have a parametrization but how do i use it? The problem is from a magazine at 8 th grade and i don't think the solution is supposed to use the general solution.

Actually the general solution of $q^2+p^2+k^2=r^2$ is given by: $ q=a^2+b^2-c^2-d^2$

$p=2(mq+np)$

$k=2(nq-mp)$

$r=a^2+b^2+c^2+d^2$ IT would follow that one of the numbers q or p is even and prime, so it is 2.

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  • $\begingroup$ Ah, it might be in the Olympiad Corner $\endgroup$ – Dietrich Burde Mar 9 at 19:53
  • $\begingroup$ @amarius8312 I have noticed that you haven't accepted any of the 14 questions you've made on MSE. Please consider accepting the answers you like the most ;) PS: You get 2 extra reputation points whenever you accept an answer $\endgroup$ – Dr. Mathva Mar 29 at 15:05
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Hint

All prime numbers $p\geq5$ satisfy $p\equiv \pm 1\mod 6$

Therefore, if $p,q,r\geq5$

$$r^2-q^2-p^2\equiv (\pm1)^2-(\pm1)^2-(\pm1)^2\equiv 1-1-1\equiv5\mod 6$$

Observe now, that there's no perfect square with the residue $5$ modulo 6.
This follows from the simple fact that (taking the equations modulo $6$)

$$\begin{array}l 1^2\equiv\color{red}1\\ 2^2\equiv \color{red}4\\ 3^2\equiv9\equiv\color{red}3\\4^2\equiv16\equiv \color{red}4\\5^2\equiv25\equiv \color{red}1 \\6^2\equiv\color{red}0\end{array}$$

Can you end it now?

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Your guess is correct: $(2,3,7)$ is the unique solution. There is a simple proof using two basic and easy to verify facts:

  • if $m$ is odd, then $m^2\equiv 1\pmod 4$;
  • if $m$ is not divisible by $3$, then $m^2\equiv 1\pmod 3$.

We proceed as follows.

Write $r^2-p^2-q^2=k^2$ with an integer $k$.

If we had $p>2$, this would imply $p^2\equiv q^2\equiv r^2\equiv 1\pmod 4$, leading to $k^2\equiv r^2-p^2-q^2\equiv 3\pmod 4$, which contradicts the first basic fact above. Thus $p=2$, whence $r^2=q^2+k^2+4$.

If we had $q>3$, this would imply $r^2\equiv q^2\equiv 1\pmod 3$ leading to $k^2\equiv 2\pmod 3$, which contradicts the second basic fact above. Thus $q=3$, and, as a result, $r^2=13+k^2$. Consequently, $(r-k)(r+k)=13$, which is only possible if $r-k=1$ and $r+k=13$. This yields $r=7$, $k=6$.

To summarize, the only solution is $p=2$, $q=3$, $r=7$.

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