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Old 2020-09-15, 08:35   #3
sweety439
 
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Nov 2016

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Many of these primes p have an easy prime of the form either (p^q-1)/(p-1) or (p^q+1)/(p+1):

(p^q-1)/(p-1): (=Rq(p))

p=263 q=5
p=311 first q is 36497 and not an easy prime
p=379 q=17
p=461 q=7
p=463 q=313 (more hard)
p=541 first q is 8951 and not an easy prime
p=751 q=967 (more hard)
p=773 q=3
p=887 q=1201 (more hard)
p=971 q=19

(p^q+1)/(p+1): (=Rq(-p))

p=263 q=13
p=311 first q is 2707 and not an easy prime
p=379 q=3
p=461 q=1889 (more hard)
p=463 q=283 (more hard)
p=541 q=3
p=751 q=23
p=773 q=7
p=887 q=1231 (more hard)
p=971 q=7

Thus, the most irregular prime p<1024 is 311

Also, for generalized Wieferich prime base p:

p=29 no known such prime
p=47 no known such prime
p=61 no known such prime
p=139 q=1822333408543
p=311 no known such prime
p=347 q=14034413930219
p=983 no known such prime

(the only B-irregular primes in this list are 311 and 347, but all of these primes except 29 and 347 are E-irregular primes, thus, 311 is the only prime in this list which is both B-irregular and E-irregular)

And more reasons:

311 is the only one small primes in the sequence: a(1)=38, a(k+1)=2*a(k)+1 (this sequence (39*2^n-1) is also the smallest k divisible by 3 without known Sophie Germain primes pair of the forms k*2^n-1 and k*2^(n+1)-1)
311 is the only one small primes in the sequence: a(1)=12, a(k+1)=3*a(k)-1
311 and 311+2 may be the only twin prime pair of the form 39*2^n+-1
(311*9^n+1)/gcd(311+1,9-1) do not have an easy prime, 311 is the first such odd number for extended Sierpinski base 9
(Let Rn(b) be the generalized repunit base b with length n = (b^n-1)/(b-1)) R311(12) has no known prime factor and be the smallest Rn(12) with no known prime factor for a long time, R311(11) and R311(10) also ever be the smallest Rn(11) and Rn(10) with no known prime factor, also the number R311(-311), it is the smallest (p^p+-1)/(p+-1) with prime p with aurifeuillean factors but both the two aurifeuillean factors have no known prime factor.
Also, 311 is much more irregular since 311^n followed by 1 is not primes for all n<=30K, it is the only such prime < 773

Last fiddled with by sweety439 on 2020-09-24 at 07:21
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