Number Theory Assignment

due Monday, May 1, 2000




  1. Exhibit Eisenstein units epsilon_{i} and Eisenstein integers xi, eta, and zeta such that

    epsilon_{1} xi^{3} + epsilon_{2} eta^{3} + epsilon_{3} zeta^{3} = 0 and xi eta zeta <> 0 .

    1. Show that any Eisenstein integer not congruent to 0 mod 2 is congruent to some power of omega mod 2, where omega = (1 + SQRT{-3})/2.

    2. Show that any Eisenstein integer not congruent to 0 mod 5 is invertible as an Eisenstein integer mod 5.

    3. Exhibit an Eisenstein integer that is invertible mod 5 but that is not congruent mod 5 to an Eisenstein unit.

  2. Let F_{p} denote the field of p elements when p is prime. Let M_{p} denote the number of points (x, y) in F_{p} \times F_{p} lying on the ``curve'' y^{2} = x^{3} + 1 .

    1. For each prime p <= 50 determine the number M_{p}. Present your results in tabular form.

    2. Are you able to prove any general statements about M_{p} ?

  3. Over the field of rational numbers find the first 6 ``multiples'' of the point (2, 3) in the arithmetic on the elliptic curve y^{2} = x^{3} + 1 .

  4. Find three pairwise non-similar right triangles with rational sides having area 6.


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