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-second Fermat number is composite

“…A glance at R22 indicates that F22 is indeed composite; in fact the table amounts to a report that all F" in the stated range are composite. The entries for R2o are in complete agreement with the report of [8] (although note that their three moduli were displayed in octal representation). where / is, say, a known small factor (not necessarily prime) and the character of G is in question.…”

“…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.…”

Discrete weighted transforms and large-integer arithmetic

“…We have not performed a full run on F22, but at this point the issue is plainly that of availability of machine cycles. We note that the composite character of F20 has been settled by direct FFT methods [21].…”

Some properties of the factors of Fermat numbers

“…Because of their quick growth, computing their factors is a challenge (only the first twelve are fully factored at the present time) and to characterize them is thus interesting. Remarkably, the so-called Pépin's test (see Weisstein [12]) has been utilized in order to prove the compositeness of 2 2 20 + 1 (see Buell & Young [14]) and of 2 2 24 + 1 (see Crandall et al [3]), although no prime factor is yet known.…”