# Goldbach's Conjecture and 1-1 correspondence [closed]

I know math only (somewhat) as a recreation, so I know this is a naive and ignorant question, but I don't have the mathematical terminology or experience to figure out why it has to be incorrect. I am not claiming a proof of Goldbach's Conjecture, just trying to determine what I'm missing in what the desired proof is supposed to show. I know the StackExchange moderation is tough and this kind of question might get closed off the bat, but I hope someone will indulge me in a little help on this one.

Here goes: If it is true that the prime numbers can be put into a 1-1 correspondence with the even numbers, and if doubling any prime number yields an even number, why doesn't that prove Goldbach's Conjecture?

Again, this sounds so trivial that of course it can't be the answer. But I'm curious about why it isn't.

Thank you for any help.

## closed as unclear what you're asking by Namaste, Xander Henderson, user284331, Saad, Chris CusterMar 31 '18 at 5:11

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• Perhaps take a look here Goldbach's Conjecture – Namaste Mar 30 '18 at 21:15
• If you read that entry, perhaps you can edit your question to more correctly represent the Conjecture, and your difficulties with it? – Namaste Mar 30 '18 at 21:25
• It's true that 1-1 correspondences between the prime numbers and the even numbers exist, but any such 1-1 correspondence won't preserve enough properties of the numbers that you can use it for anything (except some purely set theoretic things). – Henrik Mar 30 '18 at 21:32
• If this argument would work, we could also prove that every ODD number is the sum of two primes, which is false ($27$ is a counterexample) – Peter Mar 31 '18 at 16:01
• Thanks, Peter, Henrik, I think both of your comments put together explain why this wouldn't work. The correspondence alone is only part of the problem. Makes it even weirder for me as a non-mathematician, but the 27 counterexample makes it the most clear. Thanks again. – jrdevdba Apr 2 '18 at 13:52

Suppose that the prime number 7 is put in correspondence with the even number 12 (or any other even number you like).

Then it's true that $7 + 7 = 14$ is even, and hence the even number 14 is the sum of two primes, but this doesn't say anything about the even number $12$ that was the "correspondent" for 7.

• Thanks! Your answer is over my head so it seems I'll have to find a way to understand it better from a layman's viewpoint rather than bother you about re-explaining it lol. Thanks again though. – jrdevdba Mar 30 '18 at 21:13

Yes, you can make a list of the primes, and then match the list to a list of the even numbers. Suppose (just to make my argument clear) that the lists are

primes 2  3  5  7  11  13  17 ...
evens  2  4  8 10  12  13  16 ...


That happens to match them in their natural order, but any matching would do.

Now your argument doubles every prime, which will give you the list dp (for "double primes")

  dp   4  6  10  14  22  26  34 ...


All those numbers are even, but there are clearly plenty of other even numbers, like, say, $8$, not on the list dp. Those other even numbers may or may not be the sum of two primes ($8 = 3 + 5$ is). We hope they all are, which would make Goldbach prescient.

Note that carefully reading what you wrote shows that you never used anything about the list evens .

• Thanks, Ethan, I see your point about 8 being skipped. I had thought that such an apparent omission might be handled (using 1-1 correspondence) in the same way as the Hotel Problem, but I think the more relevant problem with that approach is what Henrik and Peter commented above - namely, that the mere correspondence is not enough because then "we could also prove that every ODD number is the sum of two primes, which is false (27 is a counterexample)". Thanks again! – jrdevdba Apr 2 '18 at 13:54

This has nothing to do with the question, unfortunately. The conjecture is that every even number is the sum of two primes. Your observations, though undoubtedly true, don't say anything about sums.

• Thanks! Does that mean the conjecture deals with p + p and not 2*p? So even though 2 * p could be expressed as p + p if you just use the same prime number twice, it does nothing to address the proof? – jrdevdba Mar 30 '18 at 21:11
• jrdevdba p + q where p, q are primes. In some cases, p=q, but not in all cases. – Namaste Mar 30 '18 at 21:14
• If $p$ and $q$ are odd primes, then $p+q$ is even, so if we form all such sums, we get nothing but even numbers. That's a far cry from saying that every even positive integer $>4$ occurs in the list. Yes, it is in one-to-one correspondence with the set of even numbers, but that's not enough. – saulspatz Mar 30 '18 at 21:16