On the Diophantine equation \(x(x+1) \cdots (x+n)+1= y^2\) \((17\leq n= \text{odd}\leq 27)\). (English) Zbl 1186.11014

Summary: We consider the Diophantine equation as mentioned in the title and solve it completely, i.e., show that there exist no integer solution satisfying this equation.


11D41 Higher degree equations; Fermat’s equation
11Y50 Computer solution of Diophantine equations
Full Text: DOI Euclid


[1] Abe, N.: On the Diophantine equation \(x(x+1)\cdots(x+n)+1=y^2\). Proc. Japan Acad., 76A , 16-17 (2000). · Zbl 0996.11022 · doi:10.3792/pjaa.76.16
[2] Erdös, P., and Selfridge, J. L.: The product of consecutive integers is never a power. Illinois J. Math., 19 , 292-301 (1975). · Zbl 0295.10017
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