Prime Numbers

30 Chapter 1 PRIMES!
then taken again modulo the three coprime moduli 2 36 ,2 36 − 1, 2 35 − 1to
forge a kind of “parity check” with probability of error being roughly
2 −107 . Despite the threat of machine error in a single such extensive
calculation, the agreement between the independent parties leaves little
doubt as to the composite character of F22.
(8) The character of F24—and the compositeness of the F23 cofactor—were
resolved in 1999–2000 by Crandall, Mayer, and Papadopoulos [Crandall et
al. 2003]. In this case, rigor was achieved by having (a) two independent
floating-point Pepin “wavefront” tests (by Mayer and Papadopoulos,
finishing in that order in August 1999), but also (b) a pure-integer
convolution method for deterministic checking of the Pepin squaring chain.
Again the remaining doubt as to composite character must be regarded
as minuscule. More details are discussed in Exercise 4.6.
(9) Beyond F24, everyFn through n = 32 inclusive has yielded at least one
proper factor, and all of those factors were found by trial division with
the aid of Theorem 1.3.5. (Most recently, A. Kruppa and T. Forbes found
in 2001 that 46931635677864055013377 divides F31.) The first Fermat
number of unresolved character is thus F33. Byconventional machinery
and Pepin test, the resolution of F33 would take us well beyond the next
ice age! So the need for new algorithms is as strong as can be for future
work on giant Fermat numbers.
There are many other interesting facets of Fermat numbers. There is the
challenge of finding very large composite Fn. For example, W. Keller showed
that F23471 is divisible by 5·2 23473 +1, while more recently J. Young (see [Keller
1999]) found that F213319 is divisible by 3·2 213321 +1, and even more recent is
the discovery by J. Cosgrave (who used remarkable software by Y. Gallot) that
F382447 is divisible by 3·2 382449 +1 (see Exercise 4.9). To show how hard these
investigators must have searched, the prime divisor Cosgrave found is itself
currently one of the dozen or so largest known primes. Similar efforts reported
recently in [Dubner and Gallot 2002] include K. Herranen’s generalized Fermat
prime
and S. Scott’s gargantuan prime
101830 214
+1
48594 216
+1.
A compendium of numerical results on Fermat numbers is available at [Keller
1999].
It is amusing that Fermat numbers allow still another proof of Theorem
1.1.2 that there are infinitely many primes: Since the Fermat numbers are odd
and the product of F0,F1,...,Fn−1 is Fn − 2, we immediately see that each
prime factor of Fn does not divide any earlier Fj, and so there are infinitely
many primes.
What about heuristic arguments: Can we give a suggested asymptotic
formula for the number of n ≤ x with Fn prime? If the same kind of