This is the notorious Engel problem. Dec. 1997 > The following is supposed to be one of the older Math Olympiad questions > in the (former) Soviet Union. It is supposed to have a simple > (undergrad-level) solution. Here goes: > > (a and b are positive integers): > > Let f(a,b) = (a^2+b^2) / (1+ab) > > prove that if f(a,b) is an integer, it is a square of an integer. SPOILER Hint: let m = (a^2 + b^2)/(1 + ab) with 0 < a <= b and everything integers. Show there's an integer c with 0 <= c < a and m = (a^2 + c^2)/(1 + ac). See advanced problem 6619 in the American Mathematical Monthly (solution August-September 1991) for a related problem. Robin Chapman "256 256 256. Department of Mathematics O hel, ol rite; 256; whot's University of Exeter, EX4 4QE, UK 12 tyms 256? Bugird if I no. rjc@maths.exeter.ac.uk 2 dificult 2 work out." http://www.maths.ex.ac.uk/~rjc/rjc.html Iain M. Banks - Feersum Endjinn ------------------------------------------------------------------------------- >This is an old problem from the International Mathematical Olympiad: >1988 I think. Yes, it was problem 6 in fact. IIRC, if f(a,b) is an integer then it is gcd(a,b)^2. A related problem appeared in the American Math Monthly not long after - with a,b positive integers and f(a,b) = (a^2 + b^2) / (ab + 2), prove that if f(a,b) is an integer then it is twice the square of an integer. Unfortunately more types creep in for higher values of 2 :), by which I mean using f = f_n(a,b) = (a^2 + b^2) / (ab + n) for n > 2. Modulo squares the result could always be n or (n-1)^2 + 1 (which is why n = 1,2 are special), but other sporadic values occur for certain n. I spent a while looking at these some time ago, but I don't recall what conclusions I reached. Cheers, Geoff. ------------------------------------------------------------------------------- Geoff Bailey (Fred the Wonder Worm) | Programmer by trade -- ftww@cs.su.oz.au | Gameplayer by vocation. -------------------------------------------------------------------------------