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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; 04162012 at 03:13 AM.
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