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

Richard Guy rkg at cpsc.ucalgary.ca
Sun May 16 21:13:06 CEST 2010

```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 ?

On Sun, 16 May 2010, Jonathan Post wrote:

> Substitute "natural number" for "prime" and we get a seq that njas
> rightly calls too (initially) similar to the natural numbers.
>
> Is this subset interesting?
>
> Primes that are the sum of 4 distinct primes (in at least one way, is
> it interesting to enumerate how many ways?):
>
> 17, 23, 29, 31, 37, 37, 41, 43, 47, 53, 59
>
> Examples:
>
> 17 = 2+3+5+7
> 23 = 2+3+5+13 = 2+3+7+11
> 29 = 2+3+5+19 = 2+3+7+17 = 2+3+11+13
> 31 = 2+3+7+19 = 2+5+7+17 = 2+5+11+13
> 37 = 2+3+13+19 = 2+5+7+23 = 2+5+11+19 = 2+5+13+17 = 2+7+11+17
> 41 = 2+3+5+31 = 2+3+7+29 = 2+3+13+23 = 2+5+11+23 = 2+7+13+19
> 43 = 2+3+7+31 = 2+5+7+29 = 2+5+13+23 = 2+5+17+19 = 2+7+11+23 = 2+11+13 + 17
> 47 = 2+3+5+37 = 2+3+11+31 = 2+3+13+29 = 2+5+11+29 = 2+5+17+23
> 53 = 2+3+5+43 = 2+3+7+41 = 2+3+11+37 = 2+5+17+29 = 2+7+13+31 = 2+11+17 + 23
> 59 = 2+3+7+47 = 2+3+11+43 = 2+3+13+41 = 2+5+11+41 = 2+5+23+29 =
> 2+7+13+37 = 2+7+19+31 = 2+11+17 + 29
> 61 = 2+3+13+43 = 2+5+7+47 = 2+5+11+43 = 2+5+13+41 = 2+5+17+37 =
> 2+5+23+31 = 2+7+11+41 = 2+7+23+29 = 2+11+17 + 31 = 2+11+19 + 29 =
> 2+13+17 + 29 = 2+17+19+23
> 67 = 2+5+7+53 = 2+5+13+47 = 2+5+17+43 = 2+5+19+41 = 2+5+23+37 =
> 2+7+11+47 = 2+7+17+41 = 2+11+13 + 41 = 2+11+17 + 37 = 2+13+23 + 29 =
> 2+17+19+29
> 71 = 2+3+5+61 = 2+3+7+59 = 2+5+11+53 = 2+5+17+47 = 2+5+23+41 =
> 2+7+19+43 = 2+11+17 + 41 = 2+13+19 + 37
> 73 = 2+5+7+59 = 2+5+13+53 = 2+7+11+53 = 2+7+23+41 = 2+11+17 + 43 =
> 2+11+19 + 41 = 2+13+17 + 41 = 2+19+23 + 29
> 79 = 2+5+19+53 = 2+5+23+43 = 2+7+11+59 = 2+7+17+53 = 2+7+23+47 =
> 2+11+13 + 53 = 2+13+17 + 47 = 2+13+23 + 41 = 2+17+19+41
>
> and so on (above is incomplete).
>
> Obviously, the sum of 4 odd primes is even, and thus composite, hence
> 2 must be one of these 4 distinct primes in the sums.
>
> The complement (as a subset of 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?)
>
> It is premature for me to submit these, as I have barely spent an hour
> yet on them, and thus I ask for the opinion of those who have not
> retired from programming.
>
> Thank you for your attention and consideration,
>
> Jonathan Vos Post
>
>
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>
>
>

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