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Choose any number, $x$: say, $x = 876$ (you can pick any $n$ digit number)

Now, square the number -> $876 * 876 = 767376$

But now, If I ask you the square of $ x + 1$ --> $876 + 1 = 877$. You can't solve it mentally (in most of the cases).

But instead, you can, take $x^2$ and add $ x + \{x + 1\}$ to it. Which gives --> $ x^2 + x + \{x + 1\} = 767376 + 876 + 877 = 769129 $ which interestingly is actually the square of 877!

So, this all comes down to one thing, have I discovered some new mathematical logic?

Asaf Karagila
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Anoneemus
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    You seem to have 'discovered' that $(x+1)(x+1)=x^2+2x+1$ – user2520938 Jul 31 '15 at 20:14
  • Should I be happy or sad after reading your comment? – Anoneemus Jul 31 '15 at 20:16
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    Happy: it is always fun discovering interesting things. Sad: you probably can't publish this result. – Umberto P. Jul 31 '15 at 20:17
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    @Anoneemus Happy: thinking deeply about something sometimes makes you loose sight of the bigger picture, but that doesn't mean it's not fun! Stuff happens – user2520938 Jul 31 '15 at 20:17
  • @UmbertoP. I understand that. BTW, although I'm not going to publish this anywhere, but it would be nice to know if you can tell me the places that where such things are generally published :-) – Anoneemus Jul 31 '15 at 20:20
  • @Anoneemus http://arxiv.org/ – user2520938 Jul 31 '15 at 20:21
  • @user2520938 Thanks for the link (and for the "deep thoughts" in previous msg). – Anoneemus Jul 31 '15 at 20:21
  • @Anoneemus, not to be disheartened... improved time complexities for arithmetic algorithms can sometimes be derived from elementary algebraic relations. For example Karatsuba multiplication was the first algorithm to beat "grade-school" multiplication by observing a simple algebraic identity. –  Jul 31 '15 at 20:21
  • vixra.org might be more appropriate – Umberto P. Jul 31 '15 at 20:22
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    I don't want to run down your discovery, which is legitimate mathematics, and you should be pleased with yourself for discovering it. But it is not publishable anywhere except perhaps on your blog, because it was discovered in prehistoric times. – MJD Jul 31 '15 at 20:22
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    @Anoneemus I think your insight is fun, thanks for sharing it. Seems to me it often can be faster and easier to remember to add $x$ and its successor than to remember to add $2x+1$. – Bob Pego Jul 31 '15 at 21:03

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You will notice in your general case you have $x^2+2x+1$, which is precisely the binomial expansion of $(x+1)^2$. Thus essentially you derived the logic behind the expansion of binomials.

So the result makes sense, if you want the square of some $x$ plus one you do indeed want $(x+1)^2$ or $x^2+2x+1$. This isn't a new result, but the exploration which led you to it is a very good mental exercise.

Rio Alvarado
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