[SOLVED] MA1971 Exercise Set III

1. If p and p + 2 are twin primes and p> 3, prove that 6|(p + 1). By definition, twin primes are primes that differ by exactly 2, for example 17 and 19.

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2. Show that 3 is not a rational number.
3. If F_n is the n^th Fermat number defined as F_n := 2²ⁿ + 1. Prove that F_n =

F_n²₋₁−2(F_n−2−1)². Hint: this statement can be proven with or without induction.

4. Suppose that x and y are both odd positive integers. Please show that x² + y² is not a perfect square. By definition, a perfect square is an integer n = k² for some integer k.
5. If n ∈ Z⁺, then 3|n iff three divides the sum of the digits of n.