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

Alexander P-sky apovolot at gmail.com
Mon May 17 01:12:05 CEST 2010

```What about taking extra step and making a sequence of four factor's
composite numbers, for which sum of its four factors yields a prime
number ?

On 5/16/10, William Keith <wjk26 at drexel.edu> wrote:
>
> 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

```