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Thread: Problem with Sequence of Irrational Numbers

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    Problem with Sequence of Irrational Numbers

    A hard problem in math (see details)… How do I solve…

    Let

    x = f(n)
    y = g(n)
    z = h(x,y)= 10^[(1/ y) Log (x)]

    Assume that functions f and g always return rational values for all integer n > 0.

    Problem:

    If z are irrational numbers for n = p and n = p + 1 respectively, can we get simultaneous rational values of z for n = q and n = q + 1 when p is not equal to q?

    Example

    x = f(n) = 3a^n – 1 --- can be any expression that will return rational value for x and y
    y = g(n) = 2b + 5n + 2
    z = h(x,y) = 10^[(1/y)Lox(x)]

    Let a = 2 and b = 5.

    For n = 3, we have

    x = 23
    y = 27
    z = 1.123…

    For n = 4, we have

    x = 47
    y = 32
    z = 1.128…

    Now we have two irrational values of z.

    We may conclude that we can never get rational z for this example but for some cleverly constructed expressions, we can have at least one rational number for z while the rest appear to be irrational numbers.

    So, the problem is equivalent to finding simultaneous rational values of z (for n = q and n = q + 1) after finding simultaneous irrational values (for n = p and n = p + 1).

    Help please and thanks in advance!
    Last edited by lemniscatus; 04-16-2012 at 02:13 AM.

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