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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 Fn is the nth Fermat number defined as Fn := 22n + 1. Prove that Fn =
Fn2−1−2(Fn−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.