This assignment is due next Thursday (10/8) but start it over the weekend so we can discuss it some on Tuesday.

1. Explore as thoroughly as you can the solutions (if any) to each of the Diophantine equations

   	x2 + 3y2 = n
   	x2 - 3y2 = n
   	x2 + 5y2 = n
   	x2 - 5y2 = n
   

   In particular, see if you can make a conjecture about which prime values of n allow a solution. Then see how much of that conjecture you can prove (maybe not much)!

   This is like the first question on the last assignment. In fact I built it with cut and paste. You now know that your answer should address the various implications among the three statements

   	x2 + my2 = p has a solution
   	... has a square root mod p
   	p == ... (mod ...)
   

   Prove what you can. Conjecture where you think it appropriate.

   Careful! At least one of the conjectures you might be tempted to make has a small counterexample. Look for it.

2. Explore as thoroughly as you can the solutions (if any) to the Diophantine equations

   	x2 - 3y2 = +-1
   	x2 - 5y2 = +-1
   

   You should be able to find a few small solutions by inspection and then find a pattern that allows you to prove that there are infinitely many solutions. It's harder to show that the infinite list you've found includes all the solutions.

3. Show that if the Diophantine equation

   	x2 - dy2 = +-1
   

   has one solution then it has infinitely many.

   (In fact it always does have a solution, but we're not ready to prove that yet.)

4. In class we proved that there were infinitely many primes congruent to 3 (mod 4).

   1. Explain how our argument fails when you try to use it to prove that there are infinitely many primes congruent to 1 (mod 4). (We will prove that later with a different argument.)

   2. Prove that there are infinitely many primes congruent to 5 (mod 6).

   3. Another way to state what we proved in class is that there are infinitely many primes NOT congruent to 1 mod 4. Generalize this. (See how the argument works, and generalize the argument.)

5. In class we raised the question about implications either way between the statements

   	x2 + y 2 = n  (A)
   

   and

   	-1 has a square root (mod n)        (B)
   

   when n is not prime. (We know the statements are equivalent when n is prime.)

   Right after class Professor Wortman suggested thinking about 26. 5 is a square root of -1 (mod 26) and 26 = 5² + 1², so it's not a counterexample to either implication.

   1. Prove that B implies A. Hint: glue together two of the theorems we proved in class.

   2. Prove that A does not imply B. Hint: Find a counterexample. If you can, find a counterexample that isn't a perfect square.

6. Optional. Consider the determinant |ai,j| of order 100 with ai,j = i * j. Prove that if the absolute value of each of the 100! terms in the expansion of this determinant is divided by 101 then the remainder in each case is 1. (This was Problem 1 in the afternoon session for the 17th Putnam competition, March 2, 1957.)