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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
1 year ago

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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.

Brian Yao - 1 year ago

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Indeed. Even after we found a sequence of steps that lead to a solution/proof, we have to be careful with how it is written up, and ensure that the statements flow in the direction that we desire. It is very common to end up with implication in the opposite direction of what is desired.

Calvin Lin Staff - 1 year ago

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