ASSIGNMENT 1 – ELEMENTARY NUMBER THEORY MATH 392

Eyal Goren, November 6, 1998

Answer the following questions:

1. Prove by induction the following formulae:

1^{2} + 2^{2} + 3^{2} + ⋯ n^{2} = \frac{n(n + 1)(2n + 1)}{6},

1^{3} + 2^{3} + 3^{3} + ⋯ + n^{3} = (1 + 2 + 3 + ⋯ + n)^{2}.

2. Prove, using Pascal’s triangle, that:

a) ∑_{m=0}^{n} (\begin{matrix}n \\ m \end{matrix}) = 2^{n} ; b) ∑_{m=0}^{n}(−1)^{n} (\begin{matrix}n \\ m \end{matrix}) = 0.

3) Prove, using Pascal’s triangle, that: For a fixed k the maximum of (\begin{matrix}2k \\ m \end{matrix}) is obtained for m = k, and the maximum of (\begin{matrix}2k + 1 \\ m \end{matrix}) is obtained for m = k and m = k + 1.

4) a) Prove that if n ∈ ℕ is not a square then \sqrt{n} is not rational.

b) Prove that \sqrt{2} + \sqrt{3} is not rational.

5) Do exercises (i) and (vii) from Baker’s book (p.7).

6) Pose a new question about primes, and discuss it (finding a complete solution in not required. The question should be new to you and the discussion: your own).