when will the 10M Digit prime be found?

[nomadicus]

Hmmm... I did a very simplistic log curve fit to come up with M42 being the 10M prime; I'm not a mathematician so maybe I was too simplistic. Actually what I was after was the range of candidates for P to be the next M over 10M. That's a different goal then predicting M42 .vs. M45.

Can someone explain to me why log2(log2(n)) is used to match where the current mersenne primes are located/predictied? Seems to me that using p (as in 2^P-1) based on log(p) is pretty close too. I used Excel and worked the numbers using a first order polynomial curve fit of log(p) until they seemed to line up. I don't understand the double log2()'s
(I'm not even sure I understand log2 very well).

Thanks
-=- john.

[ewmayer]

Originally Posted by nomadicus

Can someone explain to me why log2(log2(n)) is used to match where the current mersenne primes are located/predictied? Seems to me that using p (as in 2^P-1) based on log(p) is pretty close too.

As far as logarithmic plots go, logarithms to all bases (excepting 0 and 1, which are unsuitable log bases) are effectively equivalent, up to a constant factor. In other words, if you have a distribution that fits well to a straight line w.r.to the base-2 log (i.e. log2()), it'll also do so w.r.to the base-e log (i.e. log() or ln()), just with a different slope.

The reason the base-2 log is "natural" for Mersenne numbers is that taking log2 of M(p) (M(p)+1 actually, but for p large the difference rapidly becomes negligible) yields the Mersenne exponent p. Then, if we're talking about a trend like the well-known conjecture of Wagstaff (and others) that the ratio of successive M-prime exponents is on average close to a constant, another way of putting that is that plotting the logarithm (to any base) of the p's of M-primes will yield a scatter plot that is well-fit by a straight line. Since we used log2 to get p, it makes sense to also use log2 for the scatter plot, that way we don't wind up with mixed-base logarithms, as in log(log2(M(p))).