[Guest]

The infinite continued fraction: x + 1/(x+ 1/(x + 1/(x+ ....))) is denoted by F(x).

Determine all possible pair(s)(p, q) of positive integers that satisfy this equation:

2*(F(p))^-1 - (F(q))^-1 = 1

Edited August 16, 2009 by K Sengupta

[Guest]

The infinite continued fraction: x + 1/(x+ 1/(x + 1/(x+ ....))) is denoted by F(x).

Determine all possible pair(s)(p, q) of positive integers that satisfy this equation:

2*(F(p))^-1 - (F(q))^-1 = 1

Are you sure you have phrased your question correctly, KS? Please look at the analysis I have prepared.

Sengupta 2.doc

[Guest]

Are you sure you have phrased your question correctly, KS? Please look at the analysis I have prepared.

JB, I have posited my views in terms of the following comment.

Considered the analysis, but there seems to be a minor typo.

From, F(p) = (√(p² + 4) + p)/2, we have:

2*(F(p))^-1 = √(p² + 4) – p

So, apparently the given equation reduces to:

2*√(p² + 4) - √(q² + 4) – (2p - q) = 2.

[Guest]

I am sorry about my typo, KS, but my quandary still stands. Please refer to my enclosed, slightly expanded, assessment.

Sengupta 3.doc

[Guest]

I am sorry about my typo, KS, but my quandary still stands. Please refer to my enclosed, slightly expanded, assessment.

Very nicely done, JB.

You are indeed very near to the actual solution by considering almost all the possibilities.

However, these cases do not rule out the possibility that:

2*√(p² + 4) = √(q² + 4), thus cancelling out the incommensurate quantities whenever p is not equal to q.

Edited August 16, 2009 by K Sengupta

[Guest]

To paraphrase Douglas Adams from "The Hitch-hiker's Guide to the Galaxy," that thought had not even started to speculate about the merest possibility of crossing my mind. Thank you very much for the problem, KS.

p=1, q=4.