A quantitative lower bound for the greatest prime factor of (ab+1)(bc+1)(ca+1)
A quantitative primitive divisor result for points on elliptic curves
Let be an elliptic curve defined over a number field, and let be a point of infinite order. It is natural to ask how many integers fail to occur as the order of modulo a prime of . For , a quadratic twist of , and as above, we show that there is at most one such .
A quantitative version of Hurwitz' theorem on the arithmetic-geometric inequality.
A quantitative version of Runge's theorem on diophantine equations
A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.
A quantum field theoretical representation of Euler-Zagier sums.
A question of Sierpinski on triangular numbers.
A question of the distribution of points on a curve over a finite field.
A question on the discriminants of involutions of central division algebras.
A quick and dirty irreducibility test for multivariate polynomials over .
A Rademacher type formula for partitions and overpartitions.
A Rankin-Selberg integral for the adjoint representation of GL3.
A rational sixteenth power reciprocity law
A rationality condition for the existence of odd perfect numbers.
A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity
In the transformation formulas for the logarithms of the classical theta-functions, certain sums arise that are analogous to the Dedekind sums in the transformation of the logarithm of the eta-function. A new reciprocity law is established for one of these analogous sums and then applied to prove the law of quadratic reciprocity.
A Reciprocity Formula for Quadratic Forms.
A reciprocity formula for weighted quadratic partitions
A reciprocity law for K2-traces.
A reciprocity theorem and a three-term relation for generalized Dedekind-Rademacher sums
A recovery of Brouncker's proof for the quadrature continued fraction.
350 years ago in Spring of 1655 Sir William Brouncker on a request by John Wallis obtained a beautiful continued fraction for 4/π. Brouncker never published his proof. Many sources on the history of Mathematics claim that this proof was lost forever. In this paper we recover the original proof from Wallis' remarks presented in his Arithmetica Infinitorum. We show that Brouncker's and Wallis' formulas can be extended to MacLaurin's sinusoidal spirals via related Euler's products. We derive Ramanujan's...