### Calculating the first m+1 partial quotients of the n-th root of a positive rational using Lagrange's method

The method of Lagrange (1797) is used to find the the first m+1 partial quotients of (b/c)^{1/n}, b ≥ 1, c ≥ 1, b ≠ c.

We apply the algorithm for finding the continued fraction of the unique real root > 1 of a polynomial f(x) ∈ Z[x] to the case where f(x) = cx^{n} - b if b/c > 1, f(x) = bx^{n} - c if b/c < 1.
### References

*Number Theory with Computer Applications*, R. Kumanduri and C. Romero, Prentice Hall 1997, page 261
*Elements of Computer Algebra with Applications*, A.G. Akritas, Wiley 1989, 333-399.
*Art of computer programming*, volume 2, D.E. Knuth, problem 13, Ch. 4.5.3.
*Continued fractions for some algebraic numbers*, S. Lang and H. Trotter, J. für Math. **255** (1972) 112-134; Addendum 267 (1974) ibid. 219-220.
*A new proof of Vincent's theorem*, A. Alesina, M. Galuzzi, L'Enseignement Math **44** (1988), 219-256.

For a program that develops the continued fraction of *all* real roots of a suitably general polynomial, see CALC.

*Last modified 21st July 2022*

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