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Old 2020-08-17, 20:57   #6
ewmayer
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Sep 2002
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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.
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