One of the four primes must be 2. This is because the sum of four odd positive integers is
even and bigger than 2, so cannot be prime. Similarly, 2 must be used in the pair. But 2 must
not be used in the triple, otherwise its sum would be even and greater than 2.
The triple must sum to a prime that is also 2 smaller than a prime, so that the four chosen
numbers sum to a prime. The sum of the three smallest odd primes is ,
which is not prime, and so the sum of the triple must be greater than 15. The possible sums
are therefore 17, 29, 41, . . .. In order to have sum 17, one of the numbers 3, 5 or 7 must be
increased by 2. However, 3 and 5 cannot be increased by 2 as this would mean the primes in
the triple are not all different, and 7 cannot be increased by 2 as 9 is not prime. Thus the
triple cannot have sum 17. It is possible, however, to find three primes that sum to 29. For
example, 5, 7 and 17.
3 + 5 + 7 = 15
Therefore the smallest possible sum of the four primes is 29 + 2 = 31. (And an example
of four primes with all of the desired properties is ; the pair could then be
and the triple .)
{2, 5, 7, 17}
{2, 5} {5, 7, 17}