Instructions

The following five sections;

1. Direct proof
2. Proof by contrapositive
3. Proof by contradiction
4. Proof with cases
5. Proof of a biconditional

Two proofs in each section from the proposed exercises.

Proposed Exercises

1. Let a, b, c N. Prove that if a² + b² = c², then 3|ab.
2. For all x ,y ,z Z, if x³ + y³ = z³, then at least one of x ,y ,z must be a multiple of 7.
3. Does the equation n⁷ 3n⁴ 9n 7 = 0 have a solution that is a natural number? Either find a natural number solution or prove that none exists.
4. Is the following proposition true or false? Justify your conclusion.

Proposition. Let n be a natural number. If 3 does not divide (n² + 2) , then n is not a prime number or n = 3.

5.Prove that if n Z and n 2( mod 3), then there are no integers x and y such that n x² + 3y².

6.Let x R. Prove that if x⁵ + 7x³ + 5x x⁴ + x² + 8, then x 0.

7.Prove that for any integers a and b, if 4|(a² + b²), then a and b are not both odd.

8.Let p be a prime number. Prove that a² b²( mod p) if and only if a b( mod p).

9.Prove that 3 is irrational.

10.Let n be a positive integer. Prove that log₁₀(n) is irrational if and only if n > 1 and for all m Z, n 6= 10^m.

11.Let n,m Z. Show that n m is odd if and only if n + m is odd.

12.Prove that for any natural number m, there is a sequence of m consecutive natural numbers, none of which is prime.

13.Prove that for any integer n, if n is the product of 4 consecutive integers, then n + 1 is a perfect square.

We will need the following conjecture in the following two exercises:

Goldbach's Conjecture. Every even integer greater than 2 can be expressed as the sum of 2 not necessarily distinct prime numbers.

14.Prove that if Goldbach's conjecture is true, then every integer greater than 5 can beexpressed as the sum of 3 prime numbers.

15.Prove that if Goldbach's Conjecture is true, then every odd integer greater than 7 canbe written as the sum of 3 odd prime numbers.

Some proofs may satisfy more than one criteria and can be reused in multiple sections.