### Calculating the first m + 1 partial quotients of the unique positive root of a trinomial axn + bx + c

We are dealing with polyomial f(x) = axn + bx + c with integer coefficients, a > 0, c < 0. This will have a unique positive root α which we will assume is greater than 1. The method of Lagrange (1797) is used to find the the first m + 1 partial quotients of α.

We remark that Disc(f(x)) is given by

Disc(f(x)) = an-2(γnncn-1a + β(n-1)n-1bn),

where γ = (-1)n(n-1)/2 and β = (-1)(n-1)(n-2)/2.

### References

1. Number Theory with Computer Applications, R. Kumanduri and C. Romero, Prentice Hall 1997, page 261
2. Elements of Computer Algebra with Applications, A.G. Akritas, Wiley 1989, 333-399.
3. Art of computer programming, volume 2, D.E. Knuth, problem 13, Ch. 4.5.3.
4. Continued fractions for some algebraic numbers, S. Lang and H. Trotter, J. für Math. 255 (1972) 112-134; Addendum 267 (1974) ibid. 219-220.
5. A new proof of Vincent's theorem, A. Alesina, M. Galuzzi, L'Enseignement Math 44 (1988), 219-256.
6. An explanation of some exotic continued fractions found by Brillart, H.M. Stark, Computers in number theory, Academic Press 1971, 21-35.
For a program that develops the continued fraction of all real roots of a suitably general polynomial, see CALC.

Enter a (> 0):
Enter b :
Enter c (c < 0):
Enter n (1 < n ≤ 100):
Enter m (0 ≤ m < 1000):