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Deep
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 Posted: Wed Jul 09, 2008 11:27 pm Post subject: On Rationality |
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Consider the following equation under the given conditions.
R = [(a/b)k^(mk-1)]^1/k (1)
Conditions: a, b are relatively prime integers, b is even, integer m >
0, k is a prime > 7
Assertion: R cannot be be rational.
Any helpful comment upon the correctness of the assertion will be
appreciated.
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| Posted: Thu Jul 10, 2008 1:04 am Post subject: Re: On Rationality |
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On 10 Jul, 00:27, Deep <deepk...@yahoo.com> wrote:
| Quote: |
Consider the following equation under the given conditions.
R = [(a/b)k^(mk-1)]^1/k (1)
Conditions: a, b are relatively prime integers, b is even, integer m
0, k is a prime > 7
Assertion: R cannot be be rational.
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I assume you mean R = ((a/b)*k^(m*k-1))^(1/k)
Let a=11, b=2048, k=11, m=1.
Then R = 11/2, which is rational.
Let a=177147, b=53119845582848, k=11, m=2.
Then R = 33/2, which is rational.
Even more simply, let a = k * c^k, b= 2^k * d^k for suitable c and d
(c odd, d coprime to k*c).
Then R = (c/(2*d)) * k^m, which is rational.
Or let a = c^k, b= 2^k * d^k * k^(k-1) for suitable c and d (c coprime
to 2*d*k).
Then R = (c/(2*d)) * k^(m-1), which is rational. |
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