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
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>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.
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Geoff Bailey (Fred the Wonder Worm) | Programmer by trade --
ftww@cs.su.oz.au | Gameplayer by vocation.
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