I have recently, been experimenting with numerical sequences, one of which were inspired by this problem. Please view the following work which I have conducted to arrive at my proof. Images are attached.

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TopNewestNice Observation and theorem. You can establish if no has yet and you should try to have a proof and try it with large numbers because maybe they can act as counter examples.

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Thank you for the kind reply! Unfortunately, I'm not quite sure the process of a formal proof, or how to establish. It would be great if you could elaborate. Thanks!

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Well i am sorry to say that what your trying to prove is not a theorem but just common properties. This is because your stating that n^2-k=n+k which can be written as n^2-n=2k or n(n-1)/2=k. Which is nothing but mere division and nothin else.

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