Number Theory
A few examples of what you can ask Wolfram|Alpha about:
compute a prime factorization
specify a prime by its position in the sequence 2, 3, 5, ...
generate a list of primes
compute the divisors of an integer
compute a greatest common divisor
Diophantine Equations
solve a Diophantine equation
Digit Sums
sum the digits of an integer
compute the digit sum in another base
generate Pascal's triangle
compute a polygonal number
get information about a number-theoretic function
do computations with number-theoretic functions
find the continued fraction representation of a number
find continued fraction representations of a function
identify algebraic integers and units
find the minimal polynomial of an algebraic number