×

# Every Prime Except Two Can be Written as a Difference of Two Squares

$$\mathbf{Theorem:}$$ every prime except $$2$$ can be written as a difference of two square numbers

We have to find two integers $$a$$ and $$b$$ such that $$a^{2} - b^{2} = p ..... \left( i \right )$$ for a prime $$p$$

now we can rewrite equation $$\left ( i \right )$$ as,

$a^{2} - b^{2} = p$ $\Rightarrow (a - b)(a + b) = p$

now as $$p$$ is a prime number we can't write $$p$$ as a product of two integers except for $$1$$ and $$p$$ itself. from this, we get that, $a - b = 1 ....... \left( ii \right )$ $a + b = p ....... \left( iii \right)$

solving equation $$(ii)$$ and $$(iii)$$ , we get $$a = \frac{p + 1}{2}$$ and $$b = \frac{p - 1}{2}$$.

notice that $$a$$ and $$b$$ are integers if $$p$$ is an odd number. Since all primes except for $$2$$ are odd, our proof is complete for every odd primes.

now we have to proof for the even primes, which is only $$2$$. again notice that $$a$$ and $$b$$ can't be integer if $$p$$ is an even number. so we can't write $$p$$ as $$a^{2} - b^{2}$$ if $$p$$ is an even integer.

so from here, we can only deduce that we can write $$p$$ as $$a^{2} - b^{2}$$ for any prime except for $$2$$

$.$ $.$

a friend of mine claims my proof is incomplete but I can't find it. can you point it where?

Note by Tasmeem Reza
3 months, 3 weeks ago

Sort by:

If I'm not mistaken, your proof is backwards; that is, you started by assuming that every odd prime can be written as a difference of perfect squares, and then verifying that $$a$$ and $$b$$ are solutions. Essentially, what you have proven is that "if $$p = a^2 - b^2$$, then $$a = (p + 1)/2$$ and $$b = (p - 1)/2$$ is a solution." This is not quite the same as, though very closely related to, proving "there exist $$a$$ and $$b$$ such that $$p = a^2 - b^2$$." The quick fix to this problem would be to reverse your steps.

However, I believe you have successfully shown that 2 cannot be written in the form $$a^2 - b^2$$ by contradiction. So, I would guess that your friend was referring to the above issue. · 3 months, 3 weeks ago