The Twentieth Fermat Number is Composite

Abstract

Abstract.The twentieth Fermât number, F20 = 22 +1, has been proven composite by machine computation.The Fermât numbers are the numbers Fn = 22" + 1, orginally conjectured by Fermât to be prime for all n. In fact, only for n equal to 0 through 4 are they known to be prime, and small factors of Fg, Fn, F12, Fis, Fie, have been known for some time. As part of a long-term test of the hardware reliability of the Cray-2 supercomputer at the Supercomputing Research Center, the authors proved that F20 = 22 +1, which had been the smallest Fermât number of unknown character, is composite. The test for compositeness was the standard technique of Pépin The result of the computation on the Cray-2 has been verified by performing the same computation on a Cray X-MP belonging to Cray Research. The total computation time on the Cray-2 was about 10 CPU days; the time on the Cray X-MP was 82 hours. Both programs ran as single-processor programs on any available CPU of the respective machines; the ability of either computer to run in parallel on multiple CPU's was not used. The time needed to test Fn, for n in the range 10 through 20, is just slightly more than four times the time needed to test Fn_i: The number of multiplications doubles in incrementing n, and the time required for each multiplication doubles, being dependent almost entirely on the length of the operands. Our programs would thus determine the character of F22, which is now the smallest Fermât number of unknown character, in a little more than 16 times the time needed for our computation on i^o-The table below summarizes what is now known about the Fermât numbers for n less than or equal to 22. A status list for larger n appears in [1].This computation, roughly one million squarings modulo a one million bit number, would be impossible to do even on supercomputers without fast Fourier transform techniques for integer multiplication.Since one reason for performing this computation was to verify hardware reliablity and not to minimize the execution time, the program was written entirely in Cray Fortran and called Cray library functions for the FFT's. The program itself was very simple and only about 200 lines long, much of which was used for checkpointing and restarting the program. The program was called into execution every time the Cray-2 was restarted, and so

The twenty-fourth Fermat number is composite

“…Between the early 1980s and the present, massive compositeness tests have established that F 20 [31], F 22 [9,29], and as reported herein F 24 are all composite. Thus we now know that every F n with 5 ≤ n ≤ 32 is composite, and the smallest Fermat number of unknown status is, at the time of this writing, the 8-gigabit colossus F 33 .…”

“…For example, in 1997 Taura found a small factor of F 28 [19]. More recently (in fact during preparation of this manuscript,) A. Kruppa discovered the first known factor of F 31 , p = 46931635677864055013377, using a sieving program developed by T. Forbes. (This factor has p − 1 = 2 33 · 3 · 13 · 140091319777 and p + 1 = 2 · 7 · 3352259691276003929527, so could not have been found via the p − 1 or p + 1 factorization algorithms, which methods have only recently come into the realm of feasibility for numbers of this size, as discussed further in §5.…”

“…Regarding our computation, we paraphrase the claim of Young and Buell [31] regarding their calculation for F 20 , that this is the deepest ever performed for what (future cofactor tests aside) amounts to a "one-bit answer. "…”

The Twenty-Second Fermat Number is Composite

“…The procedure of evaluating R" has been used in previous years to prove various F" composite. In fact, F-j, F%, FXq, Fx$, FX4 , and F2o have been shown composite in this way [6,8]. Note that many Fn can be shown composite with relative ease, by the simple expedient of exhibiting a small, explicit factor.…”

The General Sinus

“…Even today no other Fermat primes have been found, although the smallest unsettled cases are F 22 , F 24 and F 26 (The primarily status of Fermat numbers as of 1983 can be found in Keller [15] and a shorter but more recent table is included in Young and Buell [16]). …”