Let d be a positive square-free integer. The Diophantine equation to be considered is
That this equation has an infinite number of solutions was conjectured by Fermat in 1657 and eventually solved by Lagrange. It seems that Pell had nothing to do with it,the error in attaching his name to it being due to Euler. For the whole story,and much more,the interested reader should consult the book by Edwards. See also Davenport,and A.Weil.
The solution to (30)depends upon the following proposition of Dirichlet and is an application of the pigeon hole principle.
Proposition 17.5.1 If is irrational then there are infinitely many rational number x/y,(x,y)=1 such that .
Proof.Partition the half-open interval[0,1)by
If denotes,as usual,the largest integer less than or equal to then the fractional part of is defined by .It lies in a unique member of the partition.Consider the fractional parts of .At least two of these must lie in the same subinterval. In other words there exist with,such that
(32)
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