Find the largest number that cannot be written
as a sum of distinct primes of the form 6*n+1.
Originally I thought this would be easy: just take Euclid's proof of the infinity of primes and use the new number produced as a multiple of all primes through the "highest" prime to show there's a higher prime. But research shows that this produce itself, plus 1, is not necessarily itself a prime. That would have been great as it would have been one more than a multiple of 2*3=6. So I can't prove that there is no highest prime of the form 6*n+1, or second, third, etc. highest.
However, a program that looks at successive primes doesn't seem to run out of those that are one more than a multiple of six:
76473289 36
76473337 48
76473391 54
76473409 18
76473427 18
76473457 30
76473469 12
76473571 102
76473619 48
76473673 54
76473703 30
76473721 18
76473847 126
76473889 42
76473949 60
76474003 54
76474033 30
76474039 6
76474063 24
76474087 24
76474093 6
76474141 48
76474147 6
with the differences from the preceding such prime shown on the right. It doesn't look as if such primes are dying out or becoming less common.

Posted by Charlie
on 20130826 13:44:58 