Largest integer that is not the sum of two or more different primes?

Answer 1

Well, the correct answer is 11. At least according to the link I added.

Unfortunately, I can't find a proof why. Most theorems about primes run really deep, and it's often hard to find simple intuitive arguemts for questions just like this (the Goldbach conjecture is very similar to this, but has remained unsolved for several centuries.)

This question is closely related to the concept of a complete sequence, which is a sequence which 'has enough' terms so that every positive integer can be expressed as the sum of elements in the sequence, without repeating elements.

A classic proof about the prime numbers, called Bertrand's Postulate, demonstrates that there is always a Prime number between n and 2n, for any positive integer n greater than 1.

While I can't give the proof, Bertrand's postulate basically implies that there are 'enough' primes so that, if we include the number 1, the sequence of prime numbers forms a complete sequence.

Of course, for this question, 1 is not included. So that means we don't have a complete sequence. However, apparently the sequence is 'complete' if we only consider numbers greater than 11. As the sequence of primes and 1 is complete, for any number there will be a set of unique primes and maybe 1 that add to this number. What the solution to this question says is that, for any number greater than 11, if the sum of primes and possible 1 which adds up to that number does in fact contain 1, then there will exist another sum of primes which does not contain 1. Perhaps we can take away the 1 and turn a 2 into a 3, leaving the same number as the sum. Or maye take away the 1 and add 7, then find a prime number that is in the sum that, if we take away 6, gives another prime number. To be honest I'm not quite sure how to prove this final little bit either...

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Not the correct answer for the question that was asked - which specified two OR MORE primes. 11 = 2+2+7. If the Goldbach conjecture is correct the answer is 3 (nb: 1 is not a prime).

Answer 2

Goldbach said: Every even integer greater than 2 can be expressed as the sum of two primes.

Another "version" of this says every integer greater than 5 can be written as the sum of 3 primes.

However, you are just asking for the largest integer that is NOT the sum of two or more primes. The number 3 is an integer and it is the sum of 2 which is prime and 1 which is composite. The numbers 4 and 5 certainly don't work so we can stop there using the second version of Goldbach. The number 2 is 1 +1 and that works, but 3 is larger.