Exceptional units and numbers of small Mahler measure.
Experimental determination of Apéry-like identities for .
Experimental evaluation of Euler sums.
Experimental indications of three-dimensional Galois representations from the cohomology of .
Experimental results on probable primality.
Explicit 4-descents on an elliptic curve
Explicit computations of Hilbert modular forms on .
Explicit construction of integral bases of radical function fields
We give an explicit construction of an integral basis for a radical function field , where , under the assumptions and . The field discriminant of is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the -signatures of a radical function field are also discussed in this paper.
Explicit estimates on the summatory functions of the Möbius function with coprimality restrictions
We prove that for every x > q ≥ 1, and similar estimates for the Liouville function. We also give better constants when x/q is large.,
Explicit upper bounds for |L(1,χ)| when χ(3) = 0
Let χ be a primitive Dirichlet character of conductor q and denote by L(z,χ) the associated L-series. We provide an explicit upper bound for |L(1,χ)| when 3 divides q.
Extended Euclidean Algorithm and CRT Algorithm
In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based...
Extended GCD and Hermite normal form algorithms via lattice basis reduction.
Factor tables 1657–1817, with notes on the birth of number theory
The history of the construction, organisation and publication of factor tables from 1657 to 1817, in itself a fascinating story, also touches upon many topics of general interest for the history of mathematics. The considerable labour involved in constructing and correcting these tables has pushed mathematicians and calculators to organise themselves in networks. Around 1660 J. Pell was the first to motivate others to calculate a large factor table, for which he saw many applications, from Diophantine...
Factoring and testing primes in small space
We discuss how much space is sufficient to decide whether a unary given number n is a prime. We show that O(log log n) space is sufficient for a deterministic Turing machine, if it is equipped with an additional pebble movable along the input tape, and also for an alternating machine, if the space restriction applies only to its accepting computation subtrees. In other words, the language is a prime is in pebble–DSPACE(log log n) and also in accept–ASPACE(log log n). Moreover, if the given n is...
Factoring integers with large-prime variations of the quadratic sieve.
Factoring polynomials over global fields
We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.
Factoring Polynomials with Rational Coefficients.
Factorisation sur des polynômes de degré élevé à l’aide d’un monomorphisme
Factorization Methods in Cryptosystems
Fast computation of class fields given their norm group
Let be a number field containing, for some prime , the -th roots of unity. Let be a Kummer extension of degree of characterized by its modulus and its norm group. Let be the compositum of degree extensions of of conductor dividing . Using the vector-space structure of , we suggest a modification of the rnfkummer function of PARI/GP which brings the complexity of the computation of an equation of over from exponential to linear.