4
$\begingroup$

Motivated by this earlier question, I thought of this problem:

Question: For which positive integers $n$ does there exist a prime whose decimal digits sum to $n$?

We can make two "easy" observations:

  • $1$ is not the sum of the digits of a prime.
  • $6,9,12,\ldots$ cannot be the sum of the digits of a prime (by the mod 3 version of Casting Out Nines). However, $3$ is the sum of the digits of $3$.

This makes me suspect that everything else is possible. Using the GAP code below, I checked that everything possible is achieved for $n \leq 71$ (although, this is not a particularly impressive feat).

Comment: I'm aware that questions along the lines of can the primes do that? can often be hard to answer (such as Goldbach's Conjecture), but hopefully some non-trivial contribution can be made to this question, possibly along the lines of "it is true assuming Conjecture X", or possibly via an algorithm for computing these primes faster.

GAP code:

DecimalDigitSum:=function(n)   if(n<10) then return n; fi;   return (n mod 10)+DecimalDigitSum(Int(n/10)); end;;  A:=List([1..1000],i->0);; n:=1;; while(true) do   n:=NextPrimeInt(n);   s:=DecimalDigitSum(n);   if(A[s]=0) then     Print(s," is the sum of digits of the prime ",n,"\n");     A[s]:=n;   fi; od; 

The output was:

2 is the sum of digits of the prime 2 3 is the sum of digits of the prime 3 5 is the sum of digits of the prime 5 7 is the sum of digits of the prime 7 4 is the sum of digits of the prime 13 8 is the sum of digits of the prime 17 10 is the sum of digits of the prime 19 11 is the sum of digits of the prime 29 14 is the sum of digits of the prime 59 13 is the sum of digits of the prime 67 16 is the sum of digits of the prime 79 17 is the sum of digits of the prime 89 19 is the sum of digits of the prime 199 20 is the sum of digits of the prime 389 22 is the sum of digits of the prime 499 23 is the sum of digits of the prime 599 25 is the sum of digits of the prime 997 26 is the sum of digits of the prime 1889 28 is the sum of digits of the prime 1999 29 is the sum of digits of the prime 2999 31 is the sum of digits of the prime 4999 32 is the sum of digits of the prime 6899 35 is the sum of digits of the prime 8999 34 is the sum of digits of the prime 17989 37 is the sum of digits of the prime 29989 38 is the sum of digits of the prime 39989 40 is the sum of digits of the prime 49999 41 is the sum of digits of the prime 59999 43 is the sum of digits of the prime 79999 44 is the sum of digits of the prime 98999 46 is the sum of digits of the prime 199999 47 is the sum of digits of the prime 389999 49 is the sum of digits of the prime 598999 50 is the sum of digits of the prime 599999 52 is the sum of digits of the prime 799999 53 is the sum of digits of the prime 989999 55 is the sum of digits of the prime 2998999 56 is the sum of digits of the prime 2999999 58 is the sum of digits of the prime 4999999 59 is the sum of digits of the prime 6999899 61 is the sum of digits of the prime 8989999 62 is the sum of digits of the prime 9899999 64 is the sum of digits of the prime 19999999 65 is the sum of digits of the prime 29999999 67 is the sum of digits of the prime 59899999 68 is the sum of digits of the prime 59999999 71 is the sum of digits of the prime 89999999 70 is the sum of digits of the prime 189997999 

The sequence on the right is Sloane's A067523 (or A067180 if 0's are placed in the gaps), which are not particularly enlightening. The sequence on the left does not seem to be in Sloane's OEIS.

  • 2
    Another question that comes to my mind is: Is there any integer $n$ which is the digit sum of **infinitely many** distinct primes? For example if $n=2$ were to have this property, it would imply an infinitude of generalized Fermat primes of the form $10^{2^m}+1$ which goes against usual conjectures on such primes.2015-08-18

0 Answers 0