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Monday, June 20, 2016

Continued fraction expansion of the square root of n

Results from Galois imply that the continued fraction expansion of n is of the form [a0;¯a1,a2,,a2,a1,2a0] (see also Ref.[1], page 469).

In OEIS sequences A031710, A031712, A031713, A031749 and several other sequences, the least number among the periodic part of the continued fraction expansion is considered, i.e what is δ(n)min(a1,a2,,a2,a1,2a0)?

If  t is an integer that divides 2k, then it is straightforward to verify that the continued fraction expansion of (km)2+tm for m1 is equal to [km;¯2kt,2km].  This implies the following result:

If r is even, then δ((rm2)2+m)=r for all m1.
If r is odd, then δ((rm)2+2m)=r for all m1.

References:
[1] Charles Smith, A Treatise on Algebra, Fifth Edition, Macmillan and Co., 1896.

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