20201125, 14:07  #1 
Nov 2016
2^{2}×691 Posts 
Excepted number of primes of the form (k*b^n+c)/gcd(k+c,b1) 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 a_{0}+a_{1}n+a_{2}n^{2}+a_{3}n^{3}+a_{4}n^{4}+...+a_{r}n^{r} (with r>=1 is integer, all a_{i} (0<=i<=r) integers, a_{r}>=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,b1) (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^n1)/7, (1*9^n1)/8, 4*9^n1, 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^n1, (334*10^n1)/9, 14*8^n1, etc.) or partial algebraic/partial numerical covering set (e.g. 4*24^n1, 4*39^n1, (4*19^n1)/3, (343*10^n1)/9, 1369*30^n1, 2500*55^n+1, etc.)? 
20201125, 14:09  #2 
Nov 2016
2^{2}×691 Posts 
the page only writes "trivial density O(ln(N))", however, I know that the density is different by triple (k,b,c), e.g. 29*2^n1 has much low density as 17*2^n1
Last fiddled with by sweety439 on 20201125 at 14:10 
20201130, 00:44  #3 
"Sam"
Nov 2016
2×163 Posts 
Finding the expected number of primes of (k*b^n+c)/gcd(k+c,b1) shouldn't be much different than finding the expected number of primes of the form k*b^n+c. If we treat numbers of the form k*b^n+c as "random" integers, then each candidate has probability of about
If we care about the expected number of primes from to , then we should expect or equivalently (note that for small k, it is practical to ignore the term as it will not make much of a difference in calculating the probability. Therefore, it's acceptable to rewrite as So for any sequence of randomly behaving integers, we can assume that there will be primes in the sequence up to nth term, where is a constant. Without any assumptions about the sequence of integers, we can assume that since the growth rate of k*b^n+c (or (k*b^n+c)/gcd(k+c,b1) in your case), is b. However, as you mentioned, such sequences are not "completely" random. They are random enough that the prime number theorem can be used to predict their primality, but divisibility by small primes isn't as random and can easily be predicted: Once one candidate is found to be divisible by a prime p, another predictable candidate will also be divisible by p. This decreases the probability of expected primes. Sometimes though, the candidates will never be divisible by a prime p, which increases the probability of expected primes. For example, if (k*b^n+c)/gcd(k+c,b1) is always odd, we can double our constant: , and if (k*b^n+c)/gcd(k+c,b1) is coprime to 6, we can triple our constant: , and this can go on for however many primes we know won't divide any of our candidates. The most accurate way to predict primes occurring in particular sequences from the ath term to nth term, is to sieve or remove candidates that are divisible by primes up to some constant P. Suppose we have r remaining candidates. Then we should expect about 1 in every k = r/(n1) candidates left to test for primality. The growth rate, in a sense, changes from b, to b^k. The new probability that a random integer N is prime, given that it has no prime factors less than P, is about Putting two and two together, one can obtain a more accurate estimate for the expected number of primes of the form (k*b^n+c)/gcd(k+c,b1), the new constant C is: Just in case you weren't aware: The Nash value tells you how many candidates remain after sieving a certain number of terms to a certain depth. Last fiddled with by carpetpool on 20201130 at 00:46 
20201201, 10:42  #4 
Romulan Interpreter
Jun 2011
Thailand
2·3·5^{2}·61 Posts 

20201201, 16:05  #5 
Nov 2016
2^{2}·691 Posts 
No, it is just a typo, should be "expected" ....
Last fiddled with by sweety439 on 20201201 at 16:05 
20201202, 16:26  #7  
Nov 2016
2^{2}·691 Posts 
Quote:
This conjecture would imply all Sierpinski conjectures and Riesel conjectures in CRUS to every base b>=2, also the dual Sierpinski conjecture (whether 78557 is the smallest odd number k such that 2^n+k is composite for all n>=1) and the dual Riesel conjecture (whether 509203 is the smallest odd number k such that 2^nk is composite for all n>=1), and also imply there are infinitely many such primes: * Mersenne primes * Fermat primes * Generalized repunit primes (b^n1)/(b1) to every base b>=2 not of the form m^r with r>1 * Generalized negarepunit primes (b^n+1)/(b+1) to every base b>=2 not of the form m^r with odd r>1 and not of the form 4*m^4 with integer m * Generalized Fermat primes b^(2^n)+1 to every even base b>=2 not of the form m^r with odd r>1 * Generalized half Fermat primes (b^(2^n)+1)/2 to every odd base b>=2 not of the form m^r with odd r>1 * Williams primes of the 1st kind (b1)*b^n1 to every base b>=2 * Williams primes of the 2nd kind (b1)*b^n+1 to every base b>=2 (not always true if b1 is of the form m^r with odd r>1 or of the form 4*m^4 with integer m) * Williams primes of the 3rd kind (b+1)*b^n1 to every base b>=2 Last fiddled with by sweety439 on 20201202 at 16:37 

20201202, 21:19  #8 
"Sam"
Nov 2016
2·163 Posts 
Then write up a paper explaining your conjectures, and provide more evidence for your claims, like I have given or some of your own research.

20201203, 06:40  #9 
Romulan Interpreter
Jun 2011
Thailand
9150_{10} Posts 

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