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The OEIS entry for A035251 (Positive numbers of the form $x^2 - 2y^2$ with integers $x$, $y$) states that

A positive number $n$ is representable in the form $x^2 - 2y^2$ iff every prime $p \equiv 3, 5 \pmod 8$ dividing $n$ occurs to an even power.

Probably because I didn't learn math in English, I am unable to understand what it means for a prime to occur to an even power. Primes can't be even (except $2$, but it does not fulfill the $p \equiv 3, 5 \pmod 8$ requirement), nor can they be an integer raised to any power. How would you explain this concept?

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4 Answers 4

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It means that $p$ occurs in the factorisation of $n$ as $p^n$ with $n$ even. For example, $p=2$ occurs in $12=2^2\cdot 3$ with an even power, namely as $2^2$. However, $p=3$ does not occur in $12$ with an even power.

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  • $\begingroup$ Oh, how could I not think of that. Thanks $\endgroup$
    – Maya
    Aug 14, 2017 at 19:06
  • $\begingroup$ You are welcome. $\endgroup$ Aug 14, 2017 at 19:10
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$$(\forall p\in\Bbb P)\quad 2\mid\max\{k\in\Bbb N_0: p^k\mid n\}$$

($\Bbb P$ being the set of primes)

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  • $\begingroup$ Woah, it took a moment until I realised what this meant exactly, even with the concept explained by other answers. +1 for using mathematical notation though. $\endgroup$
    – Maya
    Aug 14, 2017 at 19:34
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The prime factorization of $479001600$ is $2^{10}3^5 5^2 7\cdot 11.$ The primes $2$ and $5$ occur to even powers because their exponents are even. The primes $3$, $7,$ and $11$ occur to odd powers.

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  • $\begingroup$ Now I wonder whether 479001600 has some meaning like 42... $\endgroup$
    – Maya
    Aug 14, 2017 at 19:19
  • $\begingroup$ It's $12! = 1\cdot 2 \cdot 3 \cdots 12.$ $\endgroup$
    – B. Goddard
    Aug 14, 2017 at 19:21
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Well, $2$ is even and I'm pretty sure it's prime. But then again, I'm so old that as a child I was actually taught that $1$ is prime.

Of course for a prime to be congruent to $3$ or $5$ modulo $8$, it has to be odd, which almost all odd primes are. What "even power" refers to is the parity of the exponent. So $3$ itself is a prime to an odd power ($1$), but $3 \times 3 = 3^2$ is a prime to an even power.

For example in $3^2 5^2 = 225$, we have two odd primes both of which have an exponent that is even. This number can indeed be represented as $x^2 - 2y^2$, with $x = 45$, $y = 30$ being a nontrivial solution.

Remember also that $0$ is an even number. Then, for example, we can choose to see $7$ as $3^0 5^0 7$, and we see that $5^2 - 2 \times 3^2 = 7$ indeed.

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