[seqfan] Re: Primes that are the sum of 4 distinct primes; and Primes that are not the sum of 4 distinct primes

[William Keith]

On May 16, 2010, at 3:13 PM, Richard Guy wrote:

> If a prime is to be the sum of 4 distinct primes, then
> one of them must be even, i.e., 2.  Isn't it known
> that every odd number, prime or not, is the sum of three
> primes (Vinogradov + some hard work on the small numbers) ?
> 
> Alternatively, take 3 (say) as one of the addends, and
> we are left with the Goldbach conjecture that every even
> number is the sum of two primes ?
>> 
>> 
>> Primes that are not the sum of 4 distinct primes
>> 2, 3, 5, 7, 11, 13, 19 (what's the next value, if any?)
>> 

A prime the sum of 4 distinct primes is 2 more than an odd sum of 3 primes.  The claim that every integer n>17 is the sum of at most 3 distinct primes is equivalent to the Goldbach conjecture.  The odd case, the Odd Goldbach Problem, is true for n > 10^43000.  (Via http://primes.utm.edu/notes/conjectures/ .)  We have a ways to go yet before we can check the small cases.

Cordially,
William Keith