Strong Eulerian triples

We prove that there exist infinitely many rationals a, b and c with the property that a^2-1, b^2-1, c^2-1, ab-1, ac-1 and bc-1 are all perfect squares. This provides a solution to a variant of the problem studied by Diophantus and Euler.Comment: 8 page

Elliptic Curves and Hyperdeterminants in Quantum Gravity

Hyperdeterminants are generalizations of determinants from matrices to multi-dimensional hypermatrices. They were discovered in the 19th century by Arthur Cayley but were largely ignored over a period of 100 years before once again being recognised as important in algebraic geometry, physics and number theory. It is shown that a cubic elliptic curve whose Mordell-Weil group contains a Z2 x Z2 x Z subgroup can be transformed into the degree four hyperdeterminant on a 2x2x2 hypermatrix comprising its variables and coefficients. Furthermore, a multilinear problem defined on a 2x2x2x2 hypermatrix of coefficients can be reduced to a quartic elliptic curve whose J-invariant is expressed in terms of the hypermatrix and related invariants including the degree 24 hyperdeterminant. These connections between elliptic curves and hyperdeterminants may have applications in other areas including physics.Comment: 7 page

Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces

We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a modern point of view.Comment: 11 pages, 1 figur

When are Multiples of Polygonal Numbers again Polygonal Numbers?

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second proble

Robert F. Coleman 1954-2014

Robert F. Coleman, a highly original mathematician who has had a profound influence on modern number theory and arithmetic geometry, passed away on March 24, 2014. We give an overview of his life and career, including some of his major contributions to mathematics and his role as an activist and spokesperson for people with disabilities.Comment: 14 pages. v2: Some minor typos correcte