@article{Pu2023, 
author = {Zhen Pu and Kaimin Cheng},
title = {Consecutive integers in the form        a    x    +      y    b},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {8},
pages = {17620-17630},
keywords = {Diophantine equations, consecutive integers, sum of two powers},
url = {https://www.sciopen.com/article/10.3934/math.2023899},
doi = {10.3934/math.2023899},
abstract = {Let    a  ,  b and    k be integers greater than    1. For a tuple of    k consecutive integers sorted in ascending order, denoted by        T    k  , call        T    k   a nice    k-tuple if each integer of        T    k   is a sum of two powers of the form        a    x    +      y    b   and a perfect    k-tuple if each integer of        T    k   is a sum of two perfect powers of the form        a    x    +      y    b  , respectively. Let        N    k    (  a  ,  b  ) be the number of nice    k-tuples and                      N        ~              k    (  a  ,  b  ) be the number of perfect    k-tuples. For a given    (  a  ,  b  ), it is quite interesting to find out        N    k    (  a  ,  b  ) and                      N        ~              k    (  a  ,  b  ). In 2020, Lin and Cheng obtained the formula for        N    k    (  2  ,  2  ). The main goal of this paper is to establish the formulas for        N    k    (  a  ,  b  ) and                      N        ~              k    (  a  ,  b  ). Actually, by using the method of modulo coverage together with some elementary techniques, the formulas for                      N        ~              k    (  2  ,  2  ),                      N        ~              k    (  3  ,  2  ) and        N    k    (  3  ,  2  ) are derived.}
}