KMB, 24/11/99

M1F Foundations of Analysis—Problem Sheet 8

1)

a) If 1 ≤ m ≤ N, show that N! + m is a multiple of m.

b) Show that if T = (1, 000, 001)! + 2, then none of T, T + 1, T + 2, …,T + 1, 000, 000 are prime.

c) Can gaps between consecutive primes get arbitrarily large?

2) Let a and b be positive integers which are coprime. Let n be any integer, and assume that both a and b divide n.

a) Using the Fundamental Theorem of Arithmetic, show that ab divides n.

b) Can you show that ab divides n without using the Fundamental Theorem of Arithmetic? [Hint: use the fact that there are integers λ and μ such that λa + μb = 1.]

3) a) Show that if n is an even integer, and t ≥ 1 is an integer, then n^{t} is a multiple of 2^{t}.

b) Show that if x and y are integers, then x^{8} − y^{5} can never equal 4, in the following way: Assume that there is a solution in integers x and y.

b1) If x is even, deduce that y is even and now use (a) to deduce a contradiction.

b2) If x is odd, re-arrange and use a clever factorisation to show that there is also a contradiction in this case.

4)

a) Find three positive integers a, b, c, such that there is no prime that divides all three of a, b and c, but hcf(a, b), hcf(b, c) and hcf(c, a) are all greater than one.

b) Can you find a set S = {a, b, c, d} of four distinct positive integers, such that if x and y are any two distinct elements of S, then hcf(x, y) > 1, but also, if r, s, t are any three distinct elements of S, then no prime divides all three of r, s, t? Not the best explained question of all time, but I’m sure you get my drift.

5) A typical track by German techno nutter Alec Empire might go at, say, 250 beats per minute. A typical track by ambient popsters The Orb might go at a rather more placid 35 beats per minute. Imagine putting on a 250-beats-per-minute Alec Empire track on your CD player at the same time as putting a 35-beats-per-minute Orb track on your Dad’s record player, and starting them simultaneously, so that the first “beat” of both tracks is at the same time. [For simplicity, assume that both tracks comprise entirely of beats, let’s ignore the shouting and stuff]. The resulting explosion of sound will probably be quite difficult to understand, but eventually both tracks will “beat” again simultaneously. How many beats will Alec Empire have managed to churn out when this second simultaneous beat occurs?