Primes of the form 2.3^n+1

[Dougy]

Hi guys,
Is there a conjecture that implies that there are infinitely many primes of the form ? This is Sloane's A111974.

I've found the Williams' 1972 paper "Some Prime Numbers of the Forms $2A3^n + 1$ and $2A3^n - 1$" but that doesn't seem to help.

Thanks (:

[CRGreathouse]

Not that I know about.

The heuristic density would be O(log x/log log x) with a leading coefficient of 2/ln 3 = 1.82..., but not much is really known about the density of primes in exponential sequences. We haven't even proved that there exist a, b, c such that there are infinitely many primes of the form ax^2 + bx + c.

Admittedly, Friedlander & Iwaniec (1997) and Heath-Brown (2001) give some hope toward determining the number of primes in a polynomial sequence, but we're nowhere on exponentials. The only conjecture I know of there is on Mersenne primes, which you surely know.

[Kevin]

Originally Posted by CRGreathouse

Not that I know about.

The heuristic density would be O(log x/log log x) with a leading coefficient of 2/ln 3 = 1.82..., but not much is really known about the density of primes in exponential sequences. We haven't even proved that there exist a, b, c such that there are infinitely many primes of the form ax^2 + bx + c.

Admittedly, Friedlander & Iwaniec (1997) and Heath-Brown (2001) give some hope toward determining the number of primes in a polynomial sequence, but we're nowhere on exponentials. The only conjecture I know of there is on Mersenne primes, which you surely know.

a=0, b=2, c=1

[Dougy]

Originally Posted by CRGreathouse

Not that I know about.

The heuristic density would be O(log x/log log x) with a leading coefficient of 2/ln 3 = 1.82..., but not much is really known about the density of primes in exponential sequences. We haven't even proved that there exist a, b, c such that there are infinitely many primes of the form ax^2 + bx + c.

Admittedly, Friedlander & Iwaniec (1997) and Heath-Brown (2001) give some hope toward determining the number of primes in a polynomial sequence, but we're nowhere on exponentials. The only conjecture I know of there is on Mersenne primes, which you surely know.

Thanks, actually when you compare it to say, Sierpinksi numbers, then there's no real reason to think there'd be an infinite number of primes of this form. In fact, is there any reason to suspect there would be an infinite number of Mersenne primes?

[Kevin]

Originally Posted by Dougy

Thanks, actually when you compare it to say, Sierpinksi numbers, then there's no real reason to think there'd be an infinite number of primes of this form. In fact, is there any reason to suspect there would be an infinite number of Mersenne primes?

http://primes.utm.edu/mersenne/heuristic.html

[Dougy]

Right, thanks for that. That's somewhat convincing.

[Dougy]

So here's the graph for primes of the form 2.3^n+1 using the data in Williams and Zarnke 1972.

[CRGreathouse]

Originally Posted by Dougy

Thanks, actually when you compare it to say, Sierpinksi numbers, then there's no real reason to think there'd be an infinite number of primes of this form. In fact, is there any reason to suspect there would be an infinite number of Mersenne primes?

The current conventional wisdom, AFAIK, on Sierpinski numbers is that all have a finite covering set. Or, stronger: unless there is a finite covering set, there are infinitely many primes.

So I would expect similar things for your form, at least until I knew better.

[Dougy]

For those who are interested, I found another article on this topic:

MR2076210 (2005e:11002) Bosma, Wieb Cubic reciprocity and explicit primality tests for $h·3\sp k±1$. High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, 77--89, Fields Inst. Commun., 41, Amer. Math. Soc., Providence, RI, 2004. (Reviewer: Charles Helou) 11A15 (11A51 11Y05 11Y11)