# Computational Algebra by Willem de Graaf

By Willem de Graaf

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However, when Euler worked on the equation, he confused Brouncker with Pell, and called it the Pell equation. For this reason it is still called like this, althouh Pell never worked on it. Euler came up with a rudimentary form of the continued fraction method to solve the equation. Later Lagrange proved rigorously that this method is correct. There are many interesting questions related to the Pell equation. One is how the solution (x, y) grows with d. For example, when d = 139 the smallest solution is x = 77563250, y = 6578829.

16 GHz processor, the system got out of memory (it needed more than 32GB). 3 Exercises 1. Consider the order

Set x = bi1 · · · bit e0 er and y = p02 · · · pr2 . 5. If we are not unlucky, then gcd(n, x + y) and gcd(n, x − y) are factors of n. 16 The algorithm is based on heuristic ideas; it is not guaranteed that it manages to factor n. However, we know that n is composite. Hence if we do not find the factorisation in reasonable time we can try again; for example with a larger factor base. Implementations have shown that the algorithm works well in practice. 17 (Factorisation of Fermat numbers) n The Fermat numbers are Fn = 22 + 1.