We describe how the Hardy–Ramanujan–Rademacher formula can be implemented to allow the partition function p(n) to be computed with softly optimal complexity O(n1/2+o(1)) and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate p(1019), an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of p(n), where our implementation of the Hardy–Ramanujan–Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver’s tabulation of 76 065 congruences. Supplementary materials are available with this article.

We prove that for each positive integer k in the range 2≤k≤10 and for each positive integer k≡79 (mod 120) there is a k-step Fibonacci-like sequence of composite numbers and give some examples of such sequences. This is a natural extension of a result of Graham for the Fibonacci-like sequence.

We revise the matching algorithm of Noeske (LMS J. Comput. Math. 11 (2008) 213–222) and introduce a new approach via composition series to expedite the calculations. Furthermore, we show how the matching algorithm may be applied in the more general and frequently occurring setting that we are only given subalgebras of the condensed algebras which each contain the separable algebra of one of their Wedderburn–Malcev decompositions.

We study the differential structure of the ring of modular forms for the unit group of the quaternion algebra over ℚ of discriminant 6. Using these results we give an explicit formula for Taylor expansions of the modular forms at the elliptic points. Using appropriate normalizations we show that the Taylor coefficients at the elliptic points of the generators of the ring of modular forms are all rational and 6-integral. This gives a rational structure on the ring of modular forms. We give a recursive formula for computing the Taylor coefficients of modular forms at elliptic points and, as an application, give an algorithm for computing modular polynomials.

We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational coefficients that have many integers in a single orbit. We find several minimal quadratic rational functions with eight integers in an orbit and several minimal cubic rational functions with ten integers in an orbit. We also make some elementary observations on possibilities of an analogue of Szpiro’s conjecture in a dynamical setting and on the structure of the set of minimal models for a given rational function.

Let k be a positive integer such that k≡3 mod 4, and let N be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace Sk/2(Γ0(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F∈Sk−1(Γ0(N)), which satisfies . This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.

We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic p⩾5, our algorithm has complexity involving the seventh power of the Lie rank, which is likely to be close to best possible.

Let S1=S1(v0,…,vr+1) be the space of compactly supported C0 piecewise linear functions on a mesh M of lines through ℤ2 in directions v0,…,vr+1, possibly satisfying some restrictions on the jumps of the first order derivative. A sequence ϕ=(ϕ1,…,ϕr) of elements of S1 is called a multi-box spline if every element of S1 is a finite linear combination of shifts of (the components of) ϕ. We give some examples for multi-box splines and show that they are stable. It is further shown that any multi-box spline is not always symmetric