Is this a new theorem?

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    Algebra math Theorem
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Discussion Overview

The discussion centers around the expression a-1 x a+1 = a squared - 1, exploring whether it is an established theorem or a new discovery by the original poster. The scope includes mathematical reasoning and conceptual clarification regarding the identity and its historical significance.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the expression is well-established and known as the difference of squares.
  • One participant humorously questions the original poster's understanding of mathematics, suggesting a casual exploration rather than a formal inquiry.
  • A participant notes that the result holds only in Abelian rings, indicating that it may not apply universally, such as with matrices.
  • Another participant estimates that the concept has been known for around 5,000 years, though this is challenged by a later reply suggesting that this estimate may be exaggerated.

Areas of Agreement / Disagreement

Participants generally agree that the expression is well-known, but there are differing views on its historical context and the implications of its application in different mathematical structures.

Contextual Notes

The discussion includes assumptions about the applicability of the theorem in various mathematical contexts, such as the limitation to Abelian rings and the potential inapplicability to matrices.

zuz
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a-1 x a+1= a squared-1 Is this an established theorem or have I stumbled onto something that might get my name in the math books?
 
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zuz said:
a-1 x a+1= a squared-1 Is this an established theorem or have I stumbled onto something that might get my name in the math books?
No.
 
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pinball1970 said:
No.
To which question? Or to both? :wink:
 
zuz said:
a-1 x a+1= a squared-1 Is this an established theorem or have I stumbled onto something that might get my name in the math books?
Google difference of squares
 
berkeman said:
To which question? Or to both? :wink:
Yes
 
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zuz said:
a-1 x a+1= a squared-1 Is this an established theorem or have I stumbled onto something that might get my name in the math books?
Well-established is correct. It is one of the most important basic formulas that is used in many places. Whenever you see one of the terms ##a+1\, , \,a-1\, , \,a^2-1## or similar ##a+b\, , \,a-b\, , \,a^2-b^2## then you should think about it. I would estimate that you are roundabout 5,000 years late for the textbooks.
 
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zuz said:
a-1 x a+1= a squared-1 Is this an established theorem or have I stumbled onto something that might get my name in the math books?
Ok. High school education and you have been messing about with multiplying brackets?
Just for fun?
Good on you. I STILL do not know what mathematics actually is, I have no idea.
So I was not mocking you in any way.
If you are investigating? Good on you.
 
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A caveat: The result holds only in Abelian rings. If, e.g., A,B are matrices , result won't hold. Try expanding ##(A+B)(A-B)##, to see why, where the assumption is used.
 
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fresh_42 said:
Well-established is correct. It is one of the most important basic formulas that is used in many places. Whenever you see one of the terms ##a+1\, , \,a-1\, , \,a^2-1## or similar ##a+b\, , \,a-b\, , \,a^2-b^2## then you should think about it. I would estimate that you are roundabout 5,000 years late for the textbooks.
Thank you for a reply that is straight forward. But I think 5000 years is stretching it.
 
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zuz said:
Thank you for a reply that is straight forward. But I think 5000 years is stretching it.
Maybe a little bit. The Babylonians knew Pythagoras' theorem about 3,800 years ago.
 

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