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Mission impossible II (Posted on 2013-08-26) Difficulty: 2 of 5
Find the largest number that cannot be written as a sum of distinct primes of the form 6*n+1.

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts thoughts | Comment 1 of 7

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 2013-08-26 13:44:58
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