Many of these primes p have an easy prime of the form either (p^q1)/(p1) or (p^q+1)/(p+1):
(p^q1)/(p1): (=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 Birregular primes in this list are 311 and 347, but all of these primes except 29 and 347 are Eirregular primes, thus, 311 is the only prime in this list which is both Birregular and Eirregular)
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^n1) is also the smallest k divisible by 3 without known Sophie Germain primes pair of the forms k*2^n1 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+1,91) 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^n1)/(b1)) 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 20200924 at 07:21
