By Underwood Dudley

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**Sample text**

The finite simple continued fraction a0 , a1 , . . , an has the rational value hn /kn = rn , and is called the nth convergent to ξ. 6) relate the hi and ki to the ai . For n = 0, 2, 4, . . these convergents form a monotonically sequence with ξ as a limit. Similarly, for n = 1, 3, 5, . . the convergents form a monotonically decreasing sequence tending to ξ. The denominators kn of the convergents are an increasing sequence of positive integers for n > 0. 7), we have a0 , a1 , . . = a0 , a1 , .

If μ1 = m1 + n1 D and μ2 = m2 + n2 D, then √ μ1 μ2 = (m1 m2 + Dn1 n2 ) + (m1 n2 + m2 n1 ) D and √ μ1 μ2 = (m1 m2 + Dn1 n2 ) − (m1 n2 + m2 n1 ) D √ √ = (m1 − n1 D)(m2 − n2 D) = μ1 μ2 . Remark. 4 gives another proof of the fact that N is multiplicative. Indeed, we have N(μ1 μ2 ) = (μ1 μ2 )(μ1 μ2 ) = (μ1 μ2 )(μ1 μ2 ) = (μ1 μ1 )(μ2 μ2 ) = N(μ1 )N(μ2 ). 1 History and Motivation Euler, after a cursory reading of Wallis’s Opera Mathematica, mistakenly attributed the first serious study of nontrivial solutions to equations of the form x2 − Dy2 = 1, where x = 1 and y = 0, to John Pell.

Proof. 5 for ri − ri−1 and ri − ri−2 imply that r2j < r2j+2 , r2j−1 > r2j+1 , and r2j < r2j−1 , because the ki are positive for i ≥ 0 and the ai are positive for i ≥ 1. Thus we have r0 < r2 < r4 < . . and r1 > r3 > r5 > . . To prove that r2n < r2j−1 , we put the previous results together in the form r2n < r2n+2j < r2n+2j−1 ≤ r2j−1 . The sequence r0 , r2 , r4 , . . is monotonically increasing and is bounded above by r1 , and so has a limit. Analogously, the sequence r1 , r3 , r5 , . . is monotonically decreasing and is bounded below by r0 , and so has a limit.