Research article

A note on consecutive integers of the form 2x + y2

  • Received: 23 February 2020 Accepted: 29 April 2020 Published: 14 May 2020
  • MSC : 11D61, 11D79, 11A05

  • Let $k$ be a positive integer with $k\ge 2$. Let $N_k$ denote the number of $k$ tuples of consecutive integers with each of them in the form $2^x+y^2$, where $x, y$ are nonnegative integers. In this paper, we investigate the formulas for $N_k$. Actually, by using some elementary methods, we show that $ {N_k} = \left\{ \begin{array}{l} + \infty, \; \; \; \; \; {\rm{if}}\; {\rm{2}} \le k \le 4, \\ 6, \; \; \; \; \; \; \; \; \; {\rm{if}}\; k = 5, \\ 3, \; \; \; \; \; \; \; \; \; {\rm{if}}\; k = 6, \\ 0, \; \; \; \; \; \; \; \; \; {\rm{otherwise}}. \end{array} \right. $

    Citation: Zongbing Lin, Kaimin Cheng. A note on consecutive integers of the form 2x + y2[J]. AIMS Mathematics, 2020, 5(5): 4453-4458. doi: 10.3934/math.2020285

    Related Papers:

  • Let $k$ be a positive integer with $k\ge 2$. Let $N_k$ denote the number of $k$ tuples of consecutive integers with each of them in the form $2^x+y^2$, where $x, y$ are nonnegative integers. In this paper, we investigate the formulas for $N_k$. Actually, by using some elementary methods, we show that $ {N_k} = \left\{ \begin{array}{l} + \infty, \; \; \; \; \; {\rm{if}}\; {\rm{2}} \le k \le 4, \\ 6, \; \; \; \; \; \; \; \; \; {\rm{if}}\; k = 5, \\ 3, \; \; \; \; \; \; \; \; \; {\rm{if}}\; k = 6, \\ 0, \; \; \; \; \; \; \; \; \; {\rm{otherwise}}. \end{array} \right. $


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    [3] S. Pillai, On the inequality $0 < a^x-b^y\le n$, J. India. Math. Soc., 19 (1931), 1-11.
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    [5] S. Pillai, On the equation $2^x-3^y=2^X-3^Y$, Bull. Calcutta Math. Soc., 37 (1945), 18-20.
    [6] K. Chao, On a problem of consecutive integers, J. Sichuan University (Natural Science Edition) 2 (1962), 1-6.
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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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