I was in the Naval Academy Library in Annapolis this week, and I read a
review of
Mathematics from the Birth of Numbers
by Jan Gullberg (1997). The review praised the book highly, and I found it
in that library, and I have the 1000 page book at home now. No doubt it
deserves a lot of praise. In leafing through it, however, I found on page
82 in the section on perfect numbers the following historical error:
Euclid & Nicomachus of Geresa suggested, and Euler proved in 1750 that if
M = 2^p - 1 is a Mersenne prime, then
N = M[2^(p-1)]
is a perfect number.
A. Nicomachus doesn't prove anything.
B. Euclid proved, rather than suggested, the theorem as his last propositon
in his number books (Book 9).
[Euclid would write 2^5 - 1: 1 + 2 + 4 + 8 + 16.]
C. Euler proved that every even perfect number is obtained by Euclid's method.
I hope that I typed those formulas correctly.
Best wishes from Annapolis,
Sam Kutler