View Single Post
Old 2020-09-15, 08:19   #2
sweety439
 
sweety439's Avatar
 
Nov 2016

22×32×5×13 Posts
Default

Also,

Primes p such that (p^q-1)/(p-1) is composite for all primes q<2000: 269, 281, 311, 331, 487, 499, 541, ... [set D]

Primes p such that (p^q+1)/(p+1) is composite for all primes q<2000: 53, 97, 103, 113, 311, 313, 373, 421, 433, 479, ... [set E]

There are no generalized Wieferich prime base p less than 10^12: 29, 47, 61, 139, 311, 347, 983, ... [set F]

Consider in base b, a prime p is "much more" irregular if it satisfy more of these conditions:

* p is Bernoulli-irregular (A000928)
* p is Euler-irregular (A120337)
* p^n followed by a 1 in base b (i.e. numbers of the form b*p^n+1) is composite for all small n (usually n<10000), but b*p^n+1 do not have a trivial prime factor of all n nor have a covering set of primes
* k*p+1 is composite for k = 2, 4, 8, 10, 14, 16
* (p^q-1)/(p-1) is composite for all small prime q (usually q<2000)
* (p^q+1)/(p+1) is composite for all small prime q (usually q<2000)
* there are no small generalized Wieferich prime base p (usually < 10^12)

The primes satisfying conditions 1, 2, 4 are

263, 311, 379, 461, 463, 541, 751, 773, 887, 971, 1283, 1319, 1381, 1621, 1637, 1759, 1787, ...

Which prime is much more irregular for any given base (radix)?

Consider the first such prime p=263, base 10 have a covering set of {3,11} thus not irregular, in base 2 we have 2*263^957+1 is prime thus not irregular (if our bound of exponent n is greater than 957), it also have an easy prime for bases 4, 6, 12, 14, 16, but in base 8, 8*263^n+1 is composite for n<1000000, thus 263 is much more irregular if we use octal (263 = 407 in base 8), but not much more irregular if we use other bases <= 16

Next such prime is p=311, trivial bases are 1 mod 2, 4 mod 5, or 30 mod 31, we do not consider these bases, if our bound of exponent n is 10000, 311 is much more irregular in only 2 bases for bases b<142 (the smallest base b having a covering set): 10 and 76, thus 311 is much more irregular if we use either base 10 or base 76, but not much more irregular if we use other bases <= 142

The third such prime is p=379, trivial bases are 1 mod 2, 2 mod 3, or 6 mod 7, we do not consider these bases, if our bound of exponent n is 10000, 379 is much more irregular in only 3 bases for bases b<246 (the smallest base b having a covering set): 24, 136 and 156, thus 311 is much more irregular if we use either base 24 or base 136 or base 156, but not much more irregular if we use other bases <= 246

Next such prime is p=461, which is the first "much more irregular" prime in binary: 111001101 in base 2, since 2*461^n+1 is composite for n<=400000, and base 8 already have a covering set, base 4 is a trivial base, base 6 have an easy prime with n=1

Last fiddled with by sweety439 on 2020-09-24 at 07:14
sweety439 is offline   Reply With Quote