Tackling Fermat's last problem

Pierre de Fermat (16011665), the outstanding French mathematician, has gained public recognition following the solution by Andrew Wiles of the famous problem of Fermat's "Last Theorem". Recounted in a widely acclaimed BBC Horizon programme, the related book by Simon Singh entered the best-sellers list. Today there remains only one unresolved question of Fermat's, one of such apparent simplicity that on first hearing of it you might be tempted to exclaim: "I can solve that!". However, the consensus amongst mathematicians who have occupied themselves with this problem is that it may remain forever unresolved. Fermat's question is: what is the status of the "Fermat numbers". Are they primes (a number divisible only by 1 and itself) or composites (a number that has other factors) The Fermat numbers F0, F1, F2, F3 . . . are the unending sequence of numbers: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, . . .

These numbers (formulated by Fermat in August 1640) are formed in this simple way:

starting from 2, a succession of numbers is formed by repeatedly squaring the previous one, producing: 2, 4, 16, 256, 65536, 4294967296, and 18446744073709551617

add 1 to each, forming: 3, 5, 17, 257, etc

The nth Fermat number, Fn, is given by Fn = 2 to-the-power-of 2n+1 with n = 0,1,2,3,4,5 . . . For example F3 = 2 (to-the-power-of-2) to-the-power-of-3 +1 = 21 = 257. Fermat believed every F-number to be prime, but failed to prove it.

In 1732 the renowned Euler established that F5 (4294967297) is evenly divisible by 641 and 6700417, establishing F5 as the smallest composite (not prime) Fermat number. In the intervening years no other Fermat number has been identified as being prime, and there is now a general belief (not shared by the writer) that every Fermat number from F5 onwards is composite.

F5 to F23 are composite, but F24 (5,050,446 decimal digits), requiring a 47 by 47 feet surface to write it, allowing four digits per inch, is unresolved. A team led by Dr Richard Crandall has been attempting to establish its status as prime or composite for some time.

While F24 is large, it is insignificant compared to F382447 found by me on July 24th in St Patrick's College, Drumcondra, Dublin, to be evenly divisible by 3x2 to-the-power-of 382449 + 1 (115130 digits). This almost unimaginably large number - F382447 (over 10 to-the-power-of 115136 digits) - would require a board measuring more than 10 to-the-power-of 57550 by 10 to-the-power-of 57550 light years to write out at four digits per inch.

How has this result (a new world record for a Fermat composite, surpassing Jeffrey Young's F303088 found at the computer company Silicon Graphics) been established? It has been made possible by two French contributions: a beautiful idea (1878) from to a self-taught farmer, Francois Proth (1852-79), together with the brilliant computer program (Proth.exe) of a contemporary scientist, Yves Gallot, who lives - by a lovely twist of fate - in Toulouse, where Fermat conceived the numbers now bearing his name. Gallot shares the credit for last month's breakthrough.

Dr John Cosgrave will give a public lecture "The history of Fermat numbers from August 1640" - using Maple software, and demonstrating Gallot's program - in St Patrick's College later this year. Details will be available from www.spd.dcu.ie/johnbcos. The site gives links to Gallot's program, to the dedicated compilation work of Dr Ray Ballinger of Florida University and Dr Wilfrid Keller of Hamburg University, and to notes on Proth's 1878 theorem, and to much else besides.