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Old 2020-02-19, 18:55   #1
Mar 2018

49410 Posts
Default Why primes are either of the form 8n+3 or 8n+7

2, 3, 7, 19, 67, 79, 359, 2131, 3371, 331259 are the k's (k is prime) such that pg(k) is also prime

First I note that the primes have either the form 8n+3 or 8n+7 or 8n+2 (why?)

The primes that are not of the form 8n+3, that is 2, 7 and 359 are of the form s^2-2 where s is a prime... Infact 2=2^2-2, 7=3^2-2 and 359=19^2-2

Last fiddled with by enzocreti on 2020-02-19 at 19:06
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Old 2020-02-20, 04:00   #2
jwaltos's Avatar
Apr 2012

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One of the best ways to answer this question (without posting) is to try and find the answer in a text book or peer reviewed paper. If you haven't found it after a quick search then you haven't looked hard enough.
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Old 2020-02-20, 08:46   #3
Mar 2018

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Default ....

if a prime p has the form 8s+1 or 8s+5

then 10^d*(2^p-1)+2^(p-1)-1 is divisible by 5
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