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If $p_{i}$ and $p_{j}$ are two primes of the form $4k+1$ , with $p_{j} > p_{i}$,
show that if $p_{j} \neq$ sum of two squares $p_{i}$ is also not equal to sum of two squares.

  • It is well known that primes of the form $4k+1$ can be written as a sum of two squares. However, my task is to prove this using infinite descent. So my strategy was to assume contrary and take an arbitrary prime which is not the sum of two square (say $p_{j}$) and to prove that if $p_{j}$ is not a sum of two squares, $p_{i}$ is also not a sum of two squares. By this, I will arrive at the number $5$ in the end which is the sum of two squares and hence our assumption will become false.

  • I took two cases, one was that $p_{j}$ is a sum of three squares and the other was that $p_{j}$ is a sum of four squares. In either case, my strategy was to derive a contradiction by putting $p_{i}$ a sum of two squares. Moreover, my case division exhausted all the cases since by the sum of squares theorem, a number can be either a square, sum of two squares, sum of three squares or sum of four squares, and a prime cannot be a square. However, I cannot think of a way to derive a contradiction from either case.

Any help will be appreciated.
Thanks!

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    $\begingroup$ One can use infinite descent to prove that if a prime is $4k+1$ then it's a sum of two squares, but the descent is a little more complicated than showing that if $p$ is not such a sum then neither is the next prime lower down. $\endgroup$ – Gerry Myerson Jan 16 '15 at 16:49
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    $\begingroup$ What Gerry said - just as induction doesn't always mean using $F(n-1)$ to derive $F(n)$ for some formula $F()$, infinite descent doesn't always mean 'the next smaller thing'. It just means 'if $p$ can't be written as a sum of two squares, then there is some prime $q$ with $q\lt p$ that can't be written as a sum of two squares'. $\endgroup$ – Steven Stadnicki Jan 16 '15 at 17:01
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    $\begingroup$ Trying to prove this in this way reminds me a neighbor who tried to get rid of the snow with a cigarette lighter even though he had a perfectly good, working snowblower. $\endgroup$ – Robert Soupe Jan 16 '15 at 18:25
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    $\begingroup$ @RobertSoupe Mathematics isn't just about a bunch of tools that you can use wherever they fit right. It is a flow of logical ideas and deducing one fact from another. It is interlinked and interdependent. To say that this "tool" and "trick" will work for this problem and not for that problem undermines the spirit of the subject. Hence to say that an elementary affirmative descent method will not work for my problem is incorrect. $\endgroup$ – Henry Jan 19 '15 at 18:04
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    $\begingroup$ @RobertSoupe Moreover, even if we have found a solution to a problem using advanced methods, elementary methods allow us to attain deeper concept clarity, thorough understanding and opens doors to new realms in the subject. $\endgroup$ – Henry Jan 19 '15 at 18:06

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