Prime numbers?

Question

If an even number is the sum of two odd prime numbers, can you assume that, that even number is the sum of three prime numbers? and if an even number is the sum of three prime numbers, can you assume that it is the sum of two prime numbers?

obviously this is true up to a known even number 2e, if and only if every even number before it is the sum of two prime numbers, but what about 2e+2? obviously, it's the sum of three prime numbers. I know that this is true when one of the prime numbers that represent 2e is a lower twin prime, or if e+1 is prime itself.

I did pay attention in math class, I'm a published mathematician...

and I know that the only way that 2e+2 can be represented by three primes, if and only if one or three of the primes is 2, for every even number >4

I'm also aware of it's correlation to the goldbach conjecture. I'm simply looking for insight.

Answer 1

Only in this artificial environment you describe would the first statement be true. If it is not proved, as 'Astronomer 1980' said, whether all even numbers can be written as the sum of two primes, then it is strictly speculation.

Concerning your conjecture, as to whether 2e+2 would satisfy the question, we cannot know. We know that 2e+2 would obviously be able to be written as three primes, since 2e can be written as 2 primes, but we do not know if 2e+2 itself can be written as the sum of two primes, unless the Goldbach Conjecture is provable.
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Sir, I am no mathematician, and perhaps my analysis and that of others here did not appreciate the insight the question was asking for. What seems to be true to me though, is that just because every even up to 2e fits this rule, this does not necessarily mean the rule is general. You would have to test each candidate even by finding the actual primes whose sum would give the number. That is what I was getting at.

Your first statement, of a two-prime number being broken into three primes, says in other words that a two-prime existing would imply the next lower even also can be stated as a two-prime. Your second statement, of a three-prime existing being broken into two primes, says in other words that the next even would also be a two-prime. All of this is because of the requirement that a three-prime contains the number 2. How could such relationships be proved? I certainly don't know. Maybe that is what your question was. It seems, though, you are still left with the requirement of proving that every even is the sum of two primes.

Answer 2

I don't think you can include 3 primes. Example- 5 + 7 + 11 = 23 and it NOT an even number. That would ONLY be true if 2 were the third prime number.

If you assumed knew that there were three primes that added up to an even number such as 5 +7 +2 = 14 you could not then say for sure if you only had 2 primes that it would still be even since neither 2 +5 nor 2+ 7 will give you an even number.

Answer 3

IF every even number is the sum of two odd primes, then for the reason you gave it is also the sum of three primes if one of those primes is 2. However, it has never been proven that every even number is the sum of two prime numbers; that assertion is called the Goldbach Conjecture, and though no counterexamples have ever been found it remains unproven.

Answer 4

Too bad 1 isn't considered to be prime, or your theory might work.

I'm also not sure why others are not understanding your question. It seems reasonably worded.

An even number k is the sum of two odd prime numbers (there is no claim that k is itself prime.)

Unfortunately I don't think you'll have much luck on this forum,
I would suggest you post on sci.math.num-analysis (on usenet, you can use google groups if you like.)

Answer 5

The only way 3 primes can add to an even number is if all 3 are 2s, or if 1 of them is 2. Any other combination of 3 primes will be odd since 3 odds (all primes except 2 are odd) always add to an odd.

Answer 6

pay attention in math class!