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amenezes 2020-08-16 20:51

Lucas-Lehmer Test for Fermat Numbers.
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Here is a short paper by me on the Lucas-Lehmer Test for Fermat Numbers with a short readable Theorem and an Algorithm which implements the Lucas-Lehmer Test for the Fermat type Numbers.
Please read it and give some feedback!
It is here in pdf attached!
Allan Menezes

amenezes 2020-08-16 22:22

One can fashion a LL test using same similar code of prime95 for Mersenne Numbers that for Fermat Numbers too! The Pepin's test exists for Fermat Numbers and is deterministic and polynomial. This new LL test for Fermats is also deterministic and polynomial in running time of Order O(2^2n) for Fn=2^2^n+1 and is polynomial in Computing Space as well unlike the Pepins test.
Thank you!

JeppeSN 2020-08-17 09:00

If I understand correctly, this PARI method implements it:
[CODE]LLFermat(m) = v=Mod(5,2^(2^m)+1);for(i=1,2^m-2,v=v^2-2);v==0[/CODE]
The way I understand P├ępin's test, is:
[CODE]Pepin(m) = v=Mod(3,2^(2^m)+1);for(i=1,2^m-1,v=v^2);v==-1[/CODE]
So it looks like they are equally good.


amenezes 2020-08-17 15:10

Yes they are the same time complexity. But it would be perhaps easier to modify Prime95 for the LL test rather than the Pepin test. At least thats what i hoped, So we could hunt for both Mersenne Primes and Fermat Primes. But in the algorithm for LL for Fermat number i have included a little a bit of factorization code by gcd which would make the LL longer but would also help in factoring Fermat Composites.

ATH 2020-08-17 18:50

It is not really feasible since Fermat numbers get so incredibly big very fast, much faster than Mersenne numbers.

The smallest Fermat number with status unknown is F33 = 2[SUP]2^33[/SUP] + 1 = 2[SUP]8589934592[/SUP] + 1

This is ~82 times as many digits (NOT 82 times larger) as the current wave front of GIMPS. No computer or GPU can finish this within many years, and that is just the first unknown number. The next one F34 has twice as many digits as F33 and so on for each number.

ewmayer 2020-08-17 20:57

Thanks to the OP for sharing this interesting work, even if it offers no obvious computational advantage. As ATH notes, the Fermats are so sparse that no large distributed-computing effort a la GIMPS is useful for them.

Code-wise: my Mlucas code can handle either kind of modulus - both Mersennes and Fermats share the same core complex-FFT-based-convolution main code, but have specialized routines for the FFT pass bracketing the dyadic-mul step between end of the fwd-FFT and start of the inv-FFT, as well as specialized DWT-weight/unweight and carry propagation routines. Mersennes want a real-data FFT so the dyadc-mul step needs extra work to fiddle the complex-FFT outputs to real-data form, do the dyadic-mul, then fiddle real->complex in preparation for the iFFT. That adds ~10% overhead for Mersennes vs Fermats.

The DWT+carry steps are similarly modulus-specialized because in the Fermat case we have 3 key differences vs Mersenne:

1. In the power-of-2 transform-length case we need no Mersenne-style IBDWT, because the transform length divides the exponent, i.e. we can use a fixed base (2^16 makes the most sense for double-based FFT and Fermats up to ~F35). If n = odd*2^k is not a power of 2 - which is useful for smaller Fermats because we can squeeze more than 16 bits per input word into our FFT - we can use a Mersenne-style IBDWT, but there is a simplification in that the IBDWT weights repeat with period length [odd].

2. Fermat-mod needs an acyclic convolution, which means an extra DWT layered atop any in [1] in order to achieve that.

3. As described in the famous 1994 Crandall-Fagin IBDWT paper, Fermat-mod arithmetic is most efficiently effected using a so-called "right-angle transform" FFT variant, which leads to a different way of grouping the residue digits in machine memory.

Looking ahead a few years, I've discussed the feasibility of porting my Fermat-mod custom code to Mihai Preda's (with major contributions from George Woltman) gpuOwl program with Mihai and George, and there would appear few hurdles aside from time-for-code-and-debug: gpuOwl uses the same kind of underlying complex-FFT scheme as Mlucas. Running such a code on some cutting-edge GPU of a few years hence would appear the most feasible route to doing F33, though before running a Pepin test on that monster we'd want to do some *really* deep p-1, say a stage 1 run for ~1 year on the fastest hardware available, and should that yield no factor (as we would expect) the resulting stage 1 residue could be made available for a distributed stage 2 effort, multiple volunteers doing nonoverlapping stage 2 prime ranges for, say, another year.

LaurV 2020-08-26 10:33

I downloaded the PDF and looked into it, but I stopped after the first paragraph, and I deleted it, when I read that Fermat numbers are \(F_m=2^{2^m-1}+1\). :razz:

amenezes 2020-09-26 01:55

Sorry an obvious typo in the document. I think we all know the definition of a Fermat Number Fn. If you do not here it is. A Fermat Number is Fn=2^2^n+1

chris2be8 2020-09-26 15:46

Would it work for GFNs, b^2^n+1 where b is an arbitrary base? There are enough possible GFNs that there should be some interesting cases small enough to test in a reasonable time.


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