View Single Post
Old 2020-11-25, 14:07   #1
sweety439
 
Nov 2016

2,819 Posts
Default Excepted (k*b^n+c)/gcd(k+c,b-1) prime count for n <= given number

According to the page https://oeis.org/wiki/User:Charles_R...special_primes, the excepted number of primes of the polynomial function form a0+a1n+a2n2+a3n3+a4n4+...+arnr (with r>=1 is integer, all ai (0<=i<=r) integers, ar>=1), for 1<=n<=N is (N^(1/r))/(ln(N)), if this is an irreducible polynomial over the integers, and the values of this polynomial at n = 1, 2, 3, ... are relatively prime (Bunyakovsky conjectures is that there are infinitely many n such that this polynomial produces prime values, if this is an irreducible polynomial over the integers, and the values of this polynomial at n = 1, 2, 3, ... are relatively prime)

However, for the exponential function form (k*b^n+c)/gcd(k+c,b-1) (with k>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(k,c) = 1, gcd(b,c) = 1), what is the excepted number of primes of this form for 1<=n<=N, if this form does not have algebraic covering set (e.g. 1*8^n+1, 8*27^n+1, (1*8^n-1)/7, (1*9^n-1)/8, 4*9^n-1, 2500*16^n+1, etc.) or numerical covering set (e.g. 78557*2^n+1, (11047*3^n+1)/2, (419*4^n+1)/3, 509203*2^n-1, (334*10^n-1)/9, 14*8^n-1, etc.) or partial algebraic/partial numerical covering set (e.g. 4*24^n-1, 4*39^n-1, (4*19^n-1)/3, (343*10^n-1)/9, 1369*30^n-1, 2500*55^n+1, etc.)?
sweety439 is offline   Reply With Quote