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# The cattle problem of Archimedes

## The cattle problem of Archimedes

(OP)
See http://en.wikipedia.org/wiki/Archimedes%27_cattle_...

The cattle problem is unquestionably the most famous continued fraction problem, due both to its association with Archimedes and to the huge size of the solutions. The smallest solution has hundreds of thousands of digits. Although it is likely that numbers of this size are computable on ordinary home pcs, I will not ask for the complete solution, but only for the smallest solution to the associated Pell equation

x2 - 4729494 * y2 = 1

I believe that the main reason solutions to the cattle problem are so much bigger than the smallest solution to this Pell equation is that a solution to the cattle problem must be a solution to this Pell equation, and must also be divisible by 9314. That's a killer requirement that makes the size of the solutions explode enormously.

For just the basic Pell equation, I expect the smallest solution to have dozens of digits, not hundreds of thousands. If you can't calculate the solution exactly, answers in scientific notation to a limited number of significant figures are acceptable.

A brief overview of continued fractions and how they can be used to find solutions to the Pell equation can be found at thread1229-1709803: Continued fractions - What are they? What good are they?

This question was first posed in thread1229-1710531: The cattle problem of Archimedes. Please excuse the cross-posting.

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