Re, the prime/physics post. The spectral analysis of integers is
a way forward. However, number theorists are doing it already! The
modulo operator is the spectral analysis function and whenever n == 0
mod p then p is a factor of n. We are back to the trial division
algorithm. Faster hardware is a way forward. However, can number
theorists make a contribution? I think we can.
Probability has up to recent times not been a well defined subject.
(There are many definitions of probability). The probability that a
coin you toss lands head or tails is not a half but also includes the
possibility that I intercept it in mid air and buy sweets with it.
However, the primes numbers come to our rescue again. By providing
us with a definite known sample space. Once we have a rigorous
sample space then we can define a rigorous heuristic.
I think it is time to amplify Milton's and my work in this area of a
few months back with a little experiment.
Here is the experiment. What is the largest prp that you can find of
the GFN form
12345*2^n + 1? Using the Brown-Mills `12345+1 heuristic'.
Using Paul Jobling's NewPGen, sieve out
Candidates from n = 1 to 100
Then n= 1000 to 1100
N= 2000 to 2100 etc
Then n= 10000 to 10100,
N= 11000 to 11100, etc
And so on.
Then test them for primality with Yves Gallot's Proth.
Other `12345+1' prps are of interest but are not a proper part of
This is actually a mathematical advance. A definitive heuristic for
prime searching. Courtesy of Milton Brown, yours truly and of
course Paul Joblin's NewPGen and Yves Gallot's Proth. It has 5
parameter (a,b,c,d,e) a= 12345 b=2, c= 1, d=100 e=1000. The Brown-
Mills heuristic `12345+1' is (a,b,c,d,e)=(12345,2,1,100,1000).
Testing for primes of the GFN form 12345*2^n+1 in a range of 100
The first values are n=16, (9d) and n=26, (12d) n=29 (13d).
What is the largest GFN prime/prp you can find using this heuristic?
Within a few minutes you can find, 12345*2^3010 + 1 with 911 digits!
The point is this. The Brown-Mills heuristic algorithm is 10X
faster than any current prime searching algorithm. Pretty good
bottom line, eh!