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Topic: The Pell walk (Read 2520 times) |
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Mickey1
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The Pell walk
« on: May 22nd, 2013, 3:37am » |
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Find the first non-prime x(d) in Pell’s equation using the procedure below.
For every prime number p, construct a (possibly empty) series of natural numbers using the following procedure (start with p=2): Solve Pell’s equation xx-dyy=1 for d=p-1. We note that x (and y) depend on d and rewrite the equation
x(d)^2 – d y(d)^2=1 for clarity.
If p-1 is a square (as it is for p=2) and consequently the equations will not have a solution, continue to the next higher prime p=3.
Solve Pell-s equation for d= p-1=2 and find the solution x(2)=3. Now continue to solve the equation for a new constant
d2=x(d-1)-1=2.
In this case the equation x(2)^2-2y(2)^2=1 has the solution x(2)=3. This presents us with the second reason to stop the process, circularity in the occurrence of numbers, since 2=d3=d2.
Continue to the next prime p=5 (and dismiss it on the ground that p-1 is a square).
Note that our solution for x, x(2)=3 has been a prime. The problem is to find the first non-prime x following this procedure.
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towr
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Re: The Pell walk
« Reply #1 on: May 22nd, 2013, 10:27am » |
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Pell equations do not typically have just one solution, so the process is not well-defined.
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Wikipedia, Google, Mathworld, Integer sequence DB
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Mickey1
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Re: The Pell walk
« Reply #2 on: May 23rd, 2013, 3:29pm » |
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Yes - I meant the first or lowest solution.
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