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 Mission impossible II (Posted on 2013-08-26)
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 No Rating

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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        3676473337        4876473391        5476473409        1876473427        1876473457        3076473469        1276473571        10276473619        4876473673        5476473703        3076473721        1876473847        12676473889        4276473949        6076474003        5476474033        3076474039        676474063        2476474087        2476474093        676474141        4876474147        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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