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Number Theory (MA3Z7)
Problem Sheet I
1. Prove that there are infinitely many primes of the form 4n — 1.
[Hint: for primes p1 , . . . , pk of this form, consider 4p1 . . . pk — 1.]
2. Show that, for 2n + 1 to be prime, n must be a power of 2.
[Hint: if n is not a power of 2, then n = 2rk with k > 1 odd. Now use the identity xk + 1 = (x + 1)(xk—1 — xk—2 + · · · + 1).]
[Primes of the form 22n + 1 are called Fermat Primes.]
3. Let a,n be positive integers with a, n ≥ 2. Show that, for an — 1 to be prime, we need a = 2 and n prime.
[Use the identity xn — 1 = (x — 1)(xn—1 + · · · + 1).]
[Primes of the form 2p — 1 (with p prime) are called Mersenne Primes.]
4. Prove that (a, b) = (a, c) = 1 implies (a, bc) = 1 (for a,b, c ∈ N).
5. Use the Euclidean Algorithm to compute (826,1890).