the week ahead
Kenneth A. Ribet
I've gotten three messages from people who were without access to a
copy of the textbook when they wrote to me. One got hold of a copy
soon after writing, and another paired up with another student in the
class and can see the book that way. I hope that everything gets
resolved in short order. Meanwhile, I've changed the due date for
the next homework from September 6 to September 8 so that there's
more breathing room.
If you page through the book, you'll see that the authors talk a bit
about the prime decompositions of non-zero integers and rational
numbers and explain how to view greatest common divisors and least
common multiplies in this context. They calculate the prime
decompositions of factorials and of binomial coefficients (page 23).
After presenting this material, I intend to jump forward to Erdos's
proof of "Bertrand's Postulate," a statement about prime numbers that
was stated by J. Bertrand as an unproved assumption around 1845 and
then established by Chebyshev in 1850. Erdos supplied a beautiful
alternative proof of Bertrand's postulate in 1932, when he was a19.
Erdos's proof is explained in our book starting on page 171. See
also the Wikipedia entry http://en.wikipedia.org/wiki/
Proof_of_Bertrand%27s_postulate and the discussion in "Proofs from
the Book," a wonderful compilation of extremely beautiful proofs in
mathematics. ("The book" is God's book that contains the ideal proof
of each mathematical assertion. Erdos used to refer to this book.)
Best,
Ken Ribet