Bridge to Higher Mathematics, MA 1971
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.
2. Show that √
3 is not a rational number.
3. If Fn is the nth Fermat number defined as Fn := 22
n
+ 1. Prove that Fn =
F
2
n−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 x
2 + y
2
is
not a perfect square. By definition, a perfect square is an integer n = k
2
for some
integer k.
5. If n ∈ Z
+, then 3|n iff three divides the sum of the digits of n.
