Abstract:
The aim of our project was to prove our conjecture that the product of consecutive positive integers is never a square. In our investigation, we had developed three approaches to prove it.
In the first approach, we used the fact that a number lying between two consecutive squares is never a square to prove that the product of eight consecutive integers is never a square. Then we made use of relatively prime-ness of consecutive integers to prove the rest.
In the second approach, we had used Bertrand’s Postulate Theorem to obtain a beautiful theorem that the product of consecutive positive integers is never a square if there is a prime number among them. Besides we had found some interesting results from this theorem.
When we started our project, we thought that our conjecture had not been proved. However, we found later in a website that our conjecture has already been proved by two famous mathematicians P. Erdos and J.L. Selfridge in 1939. Although our conjecture was proved, we didn’t give up but tried our best to develop our third approach.
In the third approach, we had referred to an academic journal written by P. Erdos and J.L. Selfridge and knew that the square-free parts of consecutive integers are distinct. By counting, we arrived at a necessary condition for the product not to be a square. Unluckily, we then discovered the limitations of the third approach when the number of consecutive integers is very large. It may be due to the roughness of our estimation. Although we couldn’t complete the proof of our conjecture, we all enjoyed the process of formulating conjectures and thinking new ideas of solving problems through the cooperation among our team members in the past few months.
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