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Old 2020-10-26, 12:17   #1
Nov 2016

276410 Posts
Default 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 A046146(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)
* A085398(239) = 223, and A084740(223) = 239 (also A085398(223) = 183 and A084740(183) = 223), A250201(239) = 368 and A058013(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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