Math 425 Final Assignment

in lieu of final examination

May 8, 2000

1 Directions

This problem assignment is due in the office of the Department of Mathematics & Statistics, Room ES 110, by 3:30 p.m. on Monday, May 15, 2000.

The assignment must be prepared with great neatness. Typed assignments are preferred.

Please show your work. If you employ computational assistance, indicate clearly what methods and formulas, if any, are being used.

2 Problems

1. Find all roots mod 101 of the congruence x^{2} ≡ 87.

2. Find a primitive root mod 68921.

3. Find all positive integers a and b for which

\{ \begin{matrix}a^{2} + b^{2} & = & 169 \\ ab & = & 60 \end{matrix}

4. How many solutions are there of the congruence

x^{3} ≡ x (mod 3^{k})

when k is a positive integer?

5. Let p be a prime of the form 3k − 1. Show that every integer which is invertible mod p is congruent mod p to the cube of some integer.

6. Find a 3 × 3 invertible matrix M of rational numbers such that the transformation

( \begin{matrix}x \\ y \\ z \end{matrix} ) = M ( \begin{matrix}u \\ v \\ w \end{matrix} )

transforms the homogeneous cubic polynomial x^{3} + y^{3} − z^{3} into a homogeneous cubic polynomial in u, v, and w that is in generalized Weierstrass normal form, i.e., has zero coefficients for the three monomials v^{3}, uv^{2}, and u^{2}v.

7. Interpret the conjectural criterion m = 2 n for the congruent number problem in the case of the square-free integer d = 102. Can you find a rational right triangle with area 102?

8. Find a rational right triangle of area 30 with hypotenuse other than 13.