Number Theory Assignment

due Monday, April 10, 2000

1. List all (associate classes of) Eisenstein primes that are (either integer primes or else) prime factors of integer primes p <= 50.

2. In which of the following rings is there a ``long division'' having the property that the remainder always has (non-negative) norm strictly smaller than the norm of the divisor:

   1. R = Z + ZSQRT{-2} ?

   2. R = Z + ZSQRT{-5} ?

   Respond in each case either by proving that long division is always possible or else by giving a particular case where it is not.

3. Show that 2 is never the difference of two integer cubes except for 1 and -1.

4. Let a, b be the legs of a right triangle with hypotenuse c and area d = (a b)/2. Let x, y be given by the formulas

   {	x = - a(c - a)/2 y = a^{2}(c - a)/2

   .

   1. Show that (x, y) satisfies the equation

      y^{2} = x^{3} - d^{2} x .

   2. Given d > 0 and (x, y) satisfying the foregoing equation with y > 0, can you find a corresponding right triangle with area d as above?

5. Recalling how it was proved that an odd integer prime p is the norm of a Gaussian integer whenever the congruence x^{2} + 1 = 0 (mod p) has a solution, show that an odd integer prime p is the norm of an Eisenstein integer whenever the congruence x^{2} - x + 1 = 0 (mod p) has a solution.

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