189-235A: Basic Algebra I
Due: Wednesday, September 24.
1. Page 41, 1.48, 1.49, 1.55, 1.56 and 1.57.
(Fermat's little theorem and primality testing).
a) Show that p is prime if and only if p divides the binomial
coefficient (p/k) for all p > k >0 .
b) Show that if p is prime, then
a^p= a (mod p) for all integers a. This is known as Fermat's
c) Show that if n=1729, then a^n=a (mod n)
for all a, even though
n is not prime. Hence the converse to b) is not true.
An integer which is not prime but still
satisfies a^n= a (mod n) for all a is sometimes called
a strong pseudo-prime, or a Carmichael number. It was shown only
three years ago by A. Granville
that there are infinitely many Carmichael numbers.
Incidentally, the integer 1729 is aloso:wq of Hardy's taxicab,
and Ramanujan noted that it is remarkable for other reasons as well.