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:

(a) R = Z + Z\sqrt{−2} ?

(b) R = Z + Z\sqrt{−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 = (ab)/2. Let x, y be given by the

formulas

\{ \begin{matrix}x = −a(c − a)/2 \\ y = a^{2}(c − a)/2 \end{matrix} .

(a) Show that (x, y) satisfies the equation

y^{2} = x^{3} − d^{2}x .

(b) 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.

Document network location:

DVI: http://math.albany.edu:8000/math/pers/hammond/course/mat4525/assgt/nt000331.dvi
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