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. – Achal Jain · 5 months ago

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– Refath Bari · 5 months ago

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!Log in to reply

– Achal Jain · 5 months ago

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.Log in to reply

– Refath Bari · 5 months ago

Haha, yes. I was aware of that. Take a look at the "UPDATE"Log in to reply

– Achal Jain · 5 months ago

I knew Refath. I commented because if someone is wrong it's our responsibility to correct him/her. Note-Please Remove this bullshit or someone else will.Log in to reply

– Refath Bari · 5 months ago

Haha, Alright. Cheers (?).Log in to reply