Research article
A note on consecutive integers of the form 2x + y2
-
1.
School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, P. R. China
-
2.
School of Mathematics and Information, China West Normal University, Nanchong 637009, P. R. China
-
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
-
Abstract
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.
$
References
|
[1]
|
P. Mihailescu, Primary cyclotomic units and a proof of Catalans conjecture, J. Reine Angew. Math., 572 (2004), 167-196.
|
|
[2]
|
A. Herschfeld, The equation $2^x-3^y=d$, Bull. Amer. Math. Soc., 42 (1936), 231-234.
|
|
[3]
|
S. Pillai, On the inequality $0 < a^x-b^y\le n$, J. India. Math. Soc., 19 (1931), 1-11.
|
|
[4]
|
S. Pillai, On the equation $a^x-b^y=c$, J. India. Math. Soc., 2 (1936), 119-122.
|
|
[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.
|
-
-
-
-
DownLoad: