We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly accelerated. The most promising method depends on the use of smooth numbers. As with many algorithms depending on smooth numbers, its run time can presently only conjecturally be established; giving a rigorous analysis is one of the many open problems surrounding the Pell equation.

/pdf/solving-the-pell-equation-3iud7abosa.pdf

Solving the Pell equation

We consider the solvability of the equation and generalizations, where the A{sub i} and B are given commuting matrices over an algebraic number field F. In the semigroup membership problem, the variables x{sub i} are constrained to be nonnegative integers. While this problem is NP-complete for variable k, we give a polynomial time algorithm if k is fixed. In the group membership problem, the matrices are assumed to be invertible, and the variables x{sub i} may take on negative values. In this case we give a polynomial time algorithm for variable k and give an explicit description of the set of all solutions (as an affine lattice). The special case of 1 x 1 matrices was recently solved by Guoqiang Ge; we heavily rely on his results.

Multiplicative equations over commuting matrices

https://cr.yp.to/lineartime/dcba-20040404.pdf

Factoring into coprimes in essentially linear time

kFor multiplicative representations ni=, ay' and n;=, Byi , where ai, pj are non-zero elements of some algebraic number field K and ni, mj are rational integers, we present a deterministic polynomial time algorithm that decides whether HfZl ay' equals l$=, py'. The running time of the algorithm is polynomial in the number of bits required to represent the number field K, the elements ai,pj and the integers ni, mj.

Testing Equalities of Multiplicative Representations in Polynomial Time (Extended Abstract)

For multiplicative representations /spl Pi//sub i=1//sup k//spl alpha//sub i//sup n(i)/ and /spl Pi//sub j=1//sup l//spl beta//sub j//sup m(j)/ where /spl alpha//sub i/, /spl beta//sub j/ are non-zero elements of some algebraic number field K and n/sub i/, m/sub j/ are rational integers, we present a deterministic polynomial time algorithm that decides whether /spl Pi//sub i=1//sup k//spl alpha//sub i//sup n(i)/ equals /spl Pi//sub j=1//sup l//spl beta//sub j//sup m(j)/. The running time of the algorithm is polynomial in the number of bits required to represent the number field K, the elements /spl alpha//sub i/, /spl beta//sub j/ and the integers n/sub i/, m/sub j/. >

Testing equalities of multiplicative representations in polynomial time

We present a deterministic polynomial-time algorithm that decides whether a power product IlkI yn' is a unit in the ring of integers of K , where K is a number field, yi are nonzero elements of K and ni are rational integers. The main algorithm is based on the factor refinement method for ideals, which might be of independent interest.

/pdf/recognizing-units-in-number-fields-22ujptd4qa.pdf

Guoqiang Ge

Papers

Recognizing units in number fields