Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 73
= 8. m2(m + 1)2 _ 12. m(m + 1) (2m + 1) +
4 6
6. m(m + 1) _ m = m2(2m2 - 1).
2
If m = 2k, S = 22k(22k+l - 1). The perfect numbers are of
this form for n = 2k + 1.
6. FERMAT NUMBERS
Fermat remarked that numbers of the form 2m + 1 cannot
be prime when m contains an odd factor. For
2(2k+1)d + 1 = (2" + 1) [(2")2k - (2")2.10-1 + ... - 2" + IJ.
He remarked also that if m has no odd factor, that is, if m is
itself a power of 2, then 2m + 1 is a prime. Numbers of this
form, Fn = 2 + 2n 1, are called Fermat numbers. They increase
very rapidly: if n = 0, F" = 3; if n = 1, Fn = 5; if
n = 2, F" = 17; if n = 3, F" = 257; if n = 4, F" = 65,537.
For n = 5 we obtain a 10-digit number. To us such a number
does not seem very large, and we have methods which
enable us to decide quickly whether it is prime or composite.
But in Fermat's time this investigation was difficult, and
Fermat did not carry it out. He merely stated that all numbers
of this form are prime - an assertion which was disproved
by Euler almost a century later. Euler found that
F6 = 2 32 + 1 is divisible by 641.
Here is an elegant proof of this result which may be carried
through with little numerical calculation.
641 = 625+ 16 = 5'+ 2' = (5.2 7 ) + 1.
(This is the only calculation to be verified.) Then we have,
modulo 641,
5.27 .... -1,
whence
5.28 .... -2,
2 32 .... -1, or 2 32 + 1 .... 0,
-16.232 .... 16,
modulo 641.