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Topic: Diophantus Redux (Read 411 times) |
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ThudnBlunder
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Diophantus Redux
« on: Mar 15th, 2009, 5:01pm » |
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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?
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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Immanuel_Bonfils
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Re: Diophantus Redux
« Reply #1 on: Mar 15th, 2009, 6:41pm » |
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if x=y=even, there are infinite
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Ronno
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Re: Diophantus Redux
« Reply #2 on: Mar 15th, 2009, 8:22pm » |
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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).
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| « Last Edit: Mar 15th, 2009, 8:22pm by Ronno » |
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