BdMO National Secondary 2010#6

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samiul_samin
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BdMO National Secondary 2010#6

Unread post by samiul_samin » Fri Feb 22, 2019 11:08 pm

Find ,with proof,all the perfect squares each of which is the product of four consequetive odd natural numbers.

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samiul_samin
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Re: BdMO National Secondary 2010#6

Unread post by samiul_samin » Sat Feb 23, 2019 12:57 am

There are no such perfect square.
Proof:
Let the numbers are $(k-3),(k-1),(k+1),(k+3)$

So,there product$ =k^4-10k^2+9$
This product will be a perfect square if and only if $k=0$ but then two of the numbers will be negative.
So,there are no such perfect square which is product of four consequetive odd natural numbers.
[Q.E.D]

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