# On algebraic irrationals

01 Mar 1942-Vol. 15, Iss: 3, pp 173-176

About: The article was published on 1942-03-01. It has received 2 citations till now. The article focuses on the topics: Algebraic number.

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TL;DR: In this article, it was shown that the distance between two consecutive elements in the sequence of perfect powers tends to infinity in the Diophantine equation, i.e., the distance is at most a factor of 2.

Abstract: A perfect power is a positive integer of the form $a^x$ where $a\ge 1$ and $x\ge 2$ are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given $k\ge 1$, the number of positive integer solutions $(a,\, b,\, x,\, y)$, with $x\ge 2$ and $y\ge 2$, to the Diophantine equation $a^x-b^y=k$ is finite. This conjecture amounts to saying that the distance between two consecutive elements in the sequence of perfect powers tends to infinity. After a short introduction to Pillai's work on Diophantine questions, we quote some later developments and we discuss related open problems.

18 citations

12 Feb 2023

TL;DR: In this paper , the authors present a detailed proof of a striking result of Hardy and Littlewood, whose compact proof, which delicately uses analytic continuation, has not been written freshly anywhere since its original publication.

Abstract: There is not much that can be said for all $x$ and for all $n$ about the sum \[ \sum_{k=1}^n \frac{1}{|\sin k\pi x|}. \] However, for this and similar sums, series, and products, we can establish results for almost all $x$ using the tools of continued fractions. We present in detail the appearance of these sums in the singular series for the circle method. One particular interest of the paper is the detailed proof of a striking result of Hardy and Littlewood, whose compact proof, which delicately uses analytic continuation, has not been written freshly anywhere since its original publication. This story includes various parts of late 19th century and early 20th century mathematics.

1 citations

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TL;DR: In this article, it was shown that the distance between two consecutive elements in the sequence of perfect powers tends to infinity in the Diophantine equation, i.e., the distance is at most a factor of 2.

Abstract: A perfect power is a positive integer of the form $a^x$ where $a\ge 1$ and $x\ge 2$ are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given $k\ge 1$, the number of positive integer solutions $(a,\, b,\, x,\, y)$, with $x\ge 2$ and $y\ge 2$, to the Diophantine equation $a^x-b^y=k$ is finite. This conjecture amounts to saying that the distance between two consecutive elements in the sequence of perfect powers tends to infinity. After a short introduction to Pillai's work on Diophantine questions, we quote some later developments and we discuss related open problems.

18 citations