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        riddles >> medium >> Diophantus Redux 
        
(Message started by: ThudanBlunder on Mar 15^th, 2009, 5:01pm)

Title: Diophantus Redux
Post by ThudanBlunder on Mar 15^th, 2009, 5:01pm

How many pairs of positive integer solutions (x,y) are there to the equation 1/x + 1/y = 1/n, where n is also a positive integer?

Title: Re: Diophantus Redux
Post by Immanuel_Bonfils on Mar 15^th, 2009, 6:41pm

[hide]if x=y=even, there are infinite[/hide]

Title: Re: Diophantus Redux
Post by ronnodas on Mar 15^th, 2009, 8:22pm

[hide]For given n, there is at least two solutions (n>1):

1/(n+1) + 1/n(n+1) = 1/n and 1/2n +1/2n = 1/n

And at most n solutions since the lesser of x and y has to be >n but <=2n

If p is a divisor of n, (n(p+1)/p, n(p+1)) is a solution and all solutions seem to be of this form. If this is correct, the number of solutions is equal to the number of positive divisors of n (including n and 1).[/hide]