On Diophantine Equations of the Form $(x-a_1)(x-a_2)ldots(x-a_k)+r=y^n$

Proceedings Mathematical Sciences

Manisha Kulkarni¹  B Sury² 

[1] Poornaprajna Institute of Scientific Research, Devanahalli, Bangalore 0, India$$;Statistics & Mathematics Unit, Indian Statistical Institute, th Mile, Mysore Road, Bangalore 0 0, India$$

关键词: Diophantine equations;    ErdÅ‘s–Selfridge.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Erdős and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation $x(x+1)(x+2)ldots(x+(m-1))=y^n$ has no solutions in positive integers $x,m,n$ where $m,n>1$ and $yin Q$. We consider the equation$$(x-a_1)(x-a_2)ldots(x-a_k)+r=y^n$$where $0≤ a_1 < a_2 2 and for any nonzero integer 𝑟 which is not a perfect 𝑛-th power for which the equation admits solutions, 𝑘 is bounded by an effective bound.

【 授权许可】

Unknown