EULER SQUARES AND INCOMPLETE EULER SQUARES OF EVEN DEGREES

Graeco-Latin Squares and a Mistaken Conjecture of Euler∗

Late in his long and productive career, Leonhard Euler published a hundred-page paper detailing the properties of a new mathematical structure: Graeco-Latin squares. In this paper, Euler claimed that a Graeco-Latin square of size n could never exist for any n of the form 4k+2, although he was not able to prove it. In the end, his difficulty was validated. Over a period of 200 years, more than t...

Graeco-Latin Squares and a Mistaken Conjecture of Euler

Euler Solution Using Cartesian Grid with Least Squares Technique

subdomains exist. They include methods using unstructured meshes, body-fitted curvilinear meshes, and Cartesian meshes. This paper discusses an approach that uses “gridless” or “meshless” methods to address the boundary or interface while standard structured grid methods are used everywhere else. The present method uses the Cartesian grid to specify and distribute the computational points on th...

Partitions of Even Rank and Euler Numbers

This paper is based on an hour address given at the Sixth Conference on Formal Power Series and Algebraic Combinatorics, held at DIMACS in 1994. It is written primarily for an audience of combinato-rialists. Our hope is to publicise some intriguing enumerative conjectures which arise in the study of the homology representations of the poset of (set) partitions with an even number of blocks. The...

Values of the Euler and Carmichael Functions Which Are Sums of Three Squares

Let φ denote Euler’s totient function. The frequency with which φ(n) is a perfect square has been investigated by Banks, Friedlander, Pomerance, and Shparlinski, while the frequency with which φ(n) is a sum of two squares has been studied by Banks, Luca, Saidak, and Shparlinski. Here we look at the corresponding threesquares question. We show that φ(n) is a sum of three squares precisely sevene...