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 sweety439 2020-10-26 12:17

239

Unlike the "very irregular prime" 311 -->

* 311 is Bernoulli irregular
* 311 is Euler irregular
* k*311+1 is composite for k = 2, 4, 8, 10, 14, 16
(these three properties make Fermat Last Theorem hard to proof for exponent = 311)
* The generalized repunit number (311^p-1)/(311-1) is composite for all primes p < 30000 (the first such p is 36497)
* The generalized Wagstaff number (311^p+1)/(311+1) is composite for all primes p < 2500 (the first such p is 2707)
* The generalized half Fermat number (311^(2^n)+1)/2 is composite for all n <= 17 (this is also true for many prime bases) (the first such n is not found)
* There are no Wieferich prime base 311 < 10^12 (the first such p is not found, searched up to 2*10^13, the only other two prime bases < 1000 are 47 and 983, both of them are also Euler irregular, but Bernoulli regular)
(in fact, 311 is the only prime < 2000 satisfies at least 5 of these 7 conditions)

Also ...

* 311 is one of the only 5 primes < 1000 with smallest primitive root > 13
* 311 is the only easy prime in the sequence: a(1) = 12, a(k) = 3*k-1 for k>1, the next such prime after 311 is (23*3^20823+1)/2
* There is no known prime of the form (1244)111...111, 1244 is 4*311
* There is no prime of the form (311^n)\\1 with n<=100000 (this is also true for 173, but 173 is Bernoulli regular and Euler regular, besides, 2*173+1 = 347 is prime, also base 173 has small generalized repunit prime, small generalized Wagstaff prime, and small generalized Wieferich prime

[Thus, finding all "base 311 minimal primes" is very hard]

For the prime 239:

* 239 is Bernoulli regular, Euler regular, Sophie Germain (2*239+1 is also prime), and has small generalized repunit prime, small generalized Wagstaff prime, and small generalized Wieferich prime
* 239 is the only number n such that the generalized half Fermat number (n^2+1)/2 is fourth power (this is related to the Pell number P(7) = 169 = 13^2 and the NSW number N(7) = 239)
* and thus 239 has small order 4 Wieferich prime (13), 239 is the smallest base having order 4 Wieferich prime less than itself, besides, 11 is also Wieferich prime base 239, which makes [URL="https://oeis.org/A046146"]A046146[/URL](242) cannot be 239, this is the first case such that A046146(A007947(n)) different form A046146(n) (other than the trivial n=4, only consider the n having primitive roots)
* 239 is the largest number requiring 9 cube numbers sum to it (the only other such number is 23)
* 239 is the only Sophie Germain prime p<1000 such that the generalized repunit number Rp(b) is divisible by 2p+1 for all b<=12 (also, if we only require all b<=10, then the only such prime p<1000 is 419)
* [URL="https://oeis.org/A085398"]A085398[/URL](239) = 223, and [URL="https://oeis.org/A084740"]A084740[/URL](223) = 239 (also A085398(223) = 183 and A084740(183) = 223), [URL="https://oeis.org/A250201"]A250201[/URL](239) = 368 and [URL="https://oeis.org/A058013"]A058013[/URL](368-1) = 239
* There are only two smallest n such that 239*2^n-1 is prime: 4 and 92
* The "extended Riesel conjecture" for b=8 is only 14, but if we consider the k > CK, 239 and 247 are the smallest k's without known prime (note that for 247*8^n-1, the dual form is |8^n-247|, and the value for this form for n=1 is exactly 239), 247*8^n-1 is checked to 76666 with no prime found, does any prime of the form (239*8^n-1)/7 exists? I cannot find it.
* The "extended Sierpinski conjecture" for b=239 is only 4, but if we consider the k > CK, (6*239^n+1)/7 is composite for all n<=6000, does any prime of this form exists?

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