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Let $n,k$ two integers greater than $1$, is it possible that $n(n+1)(n+2)...(n+k)$ is a square $m^2$, with $m$ an integer ?

Thanks in advance.

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    ops my fact is wrong..sorry2011-09-22

1 Answers 1

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The answer is no, it can never be a square. This problem was originally solved by Erdos in 1939. The paper can be found here.

Later, in 1975 Erdos and Selfridge improved the result and solved a longstanding conjecture which was first considered by Liouville in the 19th century, by showing that the product of two or more consecutive positive integers is never a perfect power.

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    @Srivatsan: After my initial question, Eric responded saying something to the effect of "most of the work goes into the square case", so I responded with$a$thank you and bowed out. I then saw that Eric deleted that comment, so I deleted mine. Apparently there was more to the story (and also apparently "@Jason" doesn't work once I no longer have comments in the threads). Sorry to cause confusion!2011-09-21