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Are there infinitely many even numbers that can be expressed as the sum of two primes in two different ways?

If yes, are there infinitely many even numbers that can be expressed as the sum of two primes in three different ways?

If yes, are there infinitely many even numbers that can be expressed as the sum of two primes in $n$ different ways for all $n\geq 1$?

Last but not the least, are there infinitely many even numbers that can be expressed as the sum of $m$ primes in $n$ different ways for any given $m$ and $n$, where $m,n\geq 1$?

P.S. How does one even approach such questions? I couldn't think of any theorem that is of any help.

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  • $\begingroup$ Is there even one infinite even number that can be so expressed in two different ways? Before answering that, one would want to be clear on what an "infinite even number" is. On the other hand, if you mean "Are there infinitely many even numbers that can be expressed as the sum of two primes in two differnt ways?", then you could express it that way. In standard usage in mathematics, "infinite even numbers" means "even numbers, each one of which, by itself, is infinite". If you mean "infinitely many eve numbers" then it is incorrect to express that as "infinite even numbers". $\endgroup$ Sep 20 '15 at 15:40
  • $\begingroup$ @Michael I meant infinitely many even numbers. What exactly is an infinite even number? $\endgroup$ Sep 21 '15 at 4:08
  • $\begingroup$ Exactly. ${}\qquad{}$ $\endgroup$ Sep 21 '15 at 4:35
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The answer to all but the last question is certainly YES. By the Green-Tao theorem (https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem), there are arbitarily long arithmetic progressions in primes.

Take such a progression and then the sum of the first and the last numbers, which is also the sum of the second and the second last numbers, etc.

For the last question, write the number as the sum of two numbers and then express each of them in many ways.

Note that these give a way to express in AT LEAST n ways, not necessarily EXACTLY n ways.

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  • $\begingroup$ As for even numbers that are the sum of more than two primes (i.e. $m > 2$), just take, for instance, the first two and last two in an arithmetic progression. Those have the same sum as the second, third, penultimate and "pen-penultimate" terms, and so on. $\endgroup$
    – Arthur
    Sep 20 '15 at 15:30
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There are about $\frac n{\ln n}$ primes below $n$, hence about $\frac{n^2}{2\ln^2n}$ sums of two primes below $2n$. For $n$ large enough, the quotien $\frac{n}{4\ln^2n}$ exceeds any given $m$ thus showing that there exist numbers that can be written as sum of two primes in at least $m$ ways.

When using $k$ instead of two summands, we compare $\frac{n^k}{k!\ln^k n}$ with $kn$ instead, ending at the coresponding conclusion.

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  • $\begingroup$ This used only that $\pi(n)$ is not $O(\sqrt n)$, so is applicable to many other sequences (not to squares, though) $\endgroup$ Sep 20 '15 at 15:35

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