ASSIGNMENT 6 – ELEMENTARY NUMBER THEORY MATH 392

Eyal Goren, November 17, 1998

1. Prove that a number is rational if and only if its decimal expansion is periodic from some point onward. That is, it is of the form

b_{n} ⋯ b_{0} . b_{−1} ⋯ b_{−k}a_{1} ⋯ a_{r}a_{1} ⋯ a_{r}a_{1} ⋯ a_{r} ⋯,

for some a_{i}, b_{i}.

2. a. Compute the order of 10 in the group ℤ/pℤ for the primes p = 3, 7, 11, 13, 17. b. Write down the decimal expansion of \frac{1}{3}, \frac{1}{7}, \frac{1}{11}, \frac{1}{13}, \frac{1}{17}. Phrase a conjecture. c. Prove your conjecture.

3. Prove that there are infinitely many transcendental numbers.

4. Prove that for every natural number b, the number x defined by [0, 1, b, 1, b, 1, b, 1, b, ⋯]

does not satisfy x^{2} ∈ ℚ.

5. a. Write

π = [a_{0}, a_{1}, a_{2}, ⋯].

Using a calculator, compute a_{1}, ⋯, a_{6} and the resulting approximations.

6. Use quadratic reciprocity to calculate the following

(\frac{35}{101}) , (\frac{−2}{p}) , (\frac{12}{113}) .

7. Find for which primes p we can solve the system of equations x^{2} + y^{2} = 1, 3x^{2} + 2y^{2} = 0.