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        riddles >> easy >> The Pell walk
        
(Message started by: Mickey1 on May 22^nd, 2013, 3:37am)

Title: The Pell walk
Post by Mickey1 on May 22^nd, 2013, 3:37am

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.

Title: Re: The Pell walk
Post by towr on May 22^nd, 2013, 10:27am

Pell equations do not typically have just one solution, so the process is not well-defined.

Title: Re: The Pell walk
Post by Mickey1 on May 23^rd, 2013, 3:29pm

Yes - I meant the first or lowest solution.