Re: Prime chains x-->Ax+B
- --- In firstname.lastname@example.org, "djbroadhurst" <d.broadhurst@...> wrote:
>Thanks for those refinements, David.
> --- In email@example.com,
> "mikeoakes2" <mikeoakes2@> wrote:
> > For 1<=t<=10000, there are only the 3 we know of, namely:
> > t=1,2,6.
> Mike was asking that
> N = 3^t+4
> M = (N-2)*(N-3)+3
> be simultaneously prime. The Poisson mean for
> such cases with t > x >> 1 may be estimated as
> C/(2*log(3)^2)*sum(t>x,1/t^2) =~ C/(2*log(3)^2)/x
> where C is a constant of order unity.
> To determine C we would need to take account of the discriminant
> D = -11 of the quadratic M = (N-2)*(N-3)+3. For example,
> M is always coprime to 2, 3, 7, 13, 17, 19, 29 ...
> > Predicted number, from PNT, is:
> > 2*zeta(2)/log(3)^2=(Pi/log(3))^2/3=2.72577237357
> > which agrees nicely with the experimental value of 3.
> > Exercise for the reader: derive this formula.
> To "derive" the formula we would need to make 2 rather poor choices:
> 1) set C = 4, thus ignoring the discriminant, above,
> 2) set x = 0, thus applying the PNT all the way down to N = 7.
> More meaningfully, I estimate that C =~ 5.5 and hence that
> the probabilty of a 4th case, with t > 10^4, is about
> 5.5/(2*log(3)^2)/10^4 =~ 0.00023.
As you know, my "4" came from the fact that the 3rd and 4th terms of the sequence are known to be odd; and I didn't get round to including those other factors which increase the probability of primehood by a amall multiple, analogous to what happens for AP-k's.
I have an analogous bound, considerably less than 1, for the expected number of plupluperfect maximal power chains (in my sense) of length L=6 (of which exactly one example is known).
As L increases beyond 6, the expectation is << 1, as there are more and more powers of log(2), log(3), log(5) etc. in the denominator of the expression, and also a factor (L-2)! there. I reckon the series converges.
It's relatively unusual in [prime] number theory to find a small finite set of objects, so I'm happy to proffer the following.
Definition: A "power chain" of length L is a chain
p[n+1] = A*p[n] + B, such that p[n] is prime,
for n = 1 to L, with p > p.
Definition: A "maximal power chain" is a power chain with
integers (A,B) for which there is a proof that no chain
of greater length exists for that A.
Defininition: L2(A) = (largest prime divisor of (A-1)) - 1.
Conjecture: there are only 4 maximal power chains of length >= L2(A)+2.
$$lots for anyone who can find a 5th one (not:-)
- --- In firstname.lastname@example.org,
Kevin Acres <research@...> wrote:
> x=a*x+b either is a square or has a square as a major factor.Suppose that we want to start with a square and get a square.
Then we must solve the Diophantine equation
y^2 = a*x^2 + b
For any pair (a,b), Dario will tell us all the solutions: