20090821, 14:43  #1 
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
Primes of the form 2.3^n+1
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 (: 
20090821, 15:27  #2 
Aug 2006
2×5×593 Posts 
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 HeathBrown (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. 
20090821, 20:06  #3  
Aug 2002
Ann Arbor, MI
433 Posts 
Quote:


20090821, 20:58  #4  
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
Quote:


20090821, 22:46  #5  
Aug 2002
Ann Arbor, MI
433 Posts 
Quote:


20090822, 23:22  #6 
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
Right, thanks for that. That's somewhat convincing.

20090822, 23:40  #7 
Aug 2004
Melbourne, Australia
2^{3}×19 Posts 
So here's the graph for primes of the form 2.3^n+1 using the data in Williams and Zarnke 1972.

20090823, 06:28  #8  
Aug 2006
172A_{16} Posts 
Quote:
So I would expect similar things for your form, at least until I knew better. 

20090903, 02:44  #9 
Aug 2004
Melbourne, Australia
2^{3}·19 Posts 
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, 7789, Fields Inst. Commun., 41, Amer. Math. Soc., Providence, RI, 2004. (Reviewer: Charles Helou) 11A15 (11A51 11Y05 11Y11) 
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