- State what the mathematical notion of a combination of n things taken r at a time is (look it up on the web, for example). Give a reference (such as a URL) for your answer.
- State what binomial coefficients are, what notation is used to denote them, and how they are related to the mathematical notion of combinations. Give a reference for your answer.
- State the binomial theorem. Give a reference for your answer.
- Note that 97 is prime and congruent to 1 mod 4.
Demonstrate Fermat's Christmas Theorem by
writing 97 as the sum 97 = m
^{2}+ n^{2}of the squares of two integers m and n. - Use your answer to the previous problem to solve
x
^{2}= -1 (mod 97). Give the least positive residue modulo 97 for each solution.

The other day the question came up of how to solve quadratic equations modulo numbers other than primes. We'll explore that a bit now.

- Let p and q be positive integers, and let c and m
be integers. If x=c is a solution to x
^{2}= m (mod pq), show that x=c is also a solution to both x^{2}= m (mod p) and to x^{2}= m (mod q). - Now let p and q be distinct (i.e., different)
primes and let c, d, m and n be integers
such that c
^{2}= m (mod p) and d^{2}= m (mod q). Assume x=n is a simultaneous solution tox = c (mod p) x = d (mod q)

Show that n is a solution to x^{2}= m (mod pq). [Hint: Use the given congruences to show p and q both divide n^{2}- m. Then justify why that means that pq divides n^{2}- m.]

For these last two problems, use the ideas of the preceding two problems to find all least positive residues which are solutions, or to show that there are no solutions.

- x
^{2}= -1 (mod 35) - x
^{2}= 23 (mod 77)