### Putnam 1995

Problem B4

Express (2207-1/(2207-1/(2207-1/(2207- ... ))))1/8 in the form (a + b√c)/d, where a, b, c, d are integers.

Solution

Fairly easy. There are three parts: finding x (see below), which is easy; finding x1/8 which is less easy; and noticing that we need to prove convergence (the actual proof of convergence is straightforward).

Let x = (2207-1/(2207-1/(2207-1/(2207- ... )))). Then x = 1/(2207 - x), so x2 - 2207x + 1 = 0. Evidently x is only just less than 2207, so the root is 2207/2 + √(22072 - 4)/2 = 2207/2 + 987/2 √5.

It is easy to verify that if y = (3 + √5)/2, then y8 = x.

[How do we come up with y? Well, we are effectively told to look for something of the form (a + b√5)/c. 28 = 256, 38 = 6561, so we are looking for y between 2 and 3. We end up with a factor 2 in the denominator, so the simplest choice for c is 2. a, b = 2, 1 looks as though y will be too close to 2, so we try a, b = 3, 1.]

[Or you can take the square root three times. (a + b√5)2 = (a2 + 5b2) + 2ab√5. The simplest thing is to look for a and b both half odd integers. So we look for factors of 987. We have 987 = 3.7.47. So we can try each of the pairs (3, 329), (7, 141), (21, 47). The first two obviously do not work (the squares are too large), but the third does. Similarly, for the next two square roots.]

[Or, better, use y = x1/8. Note that 2207 + 2 = 472. Start from 2207 = y8 + 1/y8. Iterate twice more to get 3 = y + 1/y. An elegant approach, for which I thank Henry Pan from Toronto, email 29 May 2001.]

[Strictly we should also establish that the continued fraction converges and that we take the larger root for x. So, define a1= 2207, an+1 = 2207 - 1/an, b1 = 2206, bn+1 = 2207 - 1/bn. We show that an > an+1 > bn+1 > bn and that an - bn tends to zero. It follows that the continued fraction converges to a value larger than 2206, which suffices.

If y = 2207 - 1/z, then y2 - 2207y + 1 = (z2 - 2207z + 1)/z2. But a12 - 2207a1 + 1 > 0, so an2 - 2207an + 1 > 0 for all n. Hence an > 2207 - 1/an = an+1. Similarly, b12 - 2207b1 + 1 < 0, so bn2 - 2207bn + 1 < 0 for all n, and hence bn < bn+1.

an+1 - bn+1 = 1/bn - 1/an = (an - bn)/(anbn. But a1 - b1 > 0 and b1 > 0, so an - bn > 0 for all n. Also an > bn ≥ b1 = 2206, so (an+1 - bn+1) < (an - bn)/106, and hence (an - bn) tends to zero.]