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I'm reading a proof in my number theory textbook that all primes of the form $p = 4k+1$ are uniquely the sum of two squares. I'm stuck right at the beginning of the proof, where they say:

To establish the assertion, suppose that $$ p = a^2 + b^2 = c^2 + d^2 $$ where $a,b,c,d$ are all positive integers. Then $$ a^2 d^2 - b^2 c^2 = p(d^2 - b^2). $$

Perhaps I'm just missing something obvious, but I can't figure out how they managed to conclude that $a^2 d^2 - b^2 c^2 = p(d^2 - b^2).$ Please advise.

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2 Answers 2

up vote 2 down vote accepted

Write the two equations: $$p=a^2+b^2$$ $$p=c^2+d^2$$

Now, multiply the first by $d^2,$ the second by $b^2,$ and subtract the second from the first.

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First,

$$p d^2 = d^2(a^2 + b^2),$$

and then,

$$p b^2 = b^2(c^2 + d^2).$$

Subtract.

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