The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is demonstrated by the factorization of the seventh Fermât number F~ and other large numbers of interest. "Quand on a à étudier un grand nombre, il faut commencer par en déterminer quelques résidus quadratiques....

Due to the isotropy of d-dimensional hyperspherical space, one expects there to exist a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami operator. The R-radius hypersphere SR with R > 0, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace’s equation on this manifold in ter...

The lexicographic power ∆Γ of chains ∆ and Γ is, roughly, the Cartesian power ∏ γ∈Γ∆, totally ordered lexicographically from the left. Here the focus is on certain powers in which either ∆ = R or Γ = R, with emphasis on when two such powers are isomorphic and on when ∆Γ is 2-homogeneous. The main results are: 1) For a countably infinite ordinal α, Rα+α ' R. 2) RR 6' RQ. 3) For ∆ a countable ord...

We note that Birch’s Theorem 1*, as well as his conjecture, which was proved in full by Helge Tverberg in 1964, fifty years ago (see [17]), and thus is now known as Tverberg’s theorem [13], are tight : This is not only evident from concrete configurations, but also from a general position argument: If (r− 1)(d+ 1) points in R in general position are partitioned into r subsets, then not even the...