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## Sum of odd number is a square, whos theorem is this?

1=1^2 1+3=2^2 1+3+5=3^2 1+3+5+7=4^2

Who's is this theorem, who is the first to realyse this? Nicomah, Aristotel, Archimedes? I need to know for sure and fast plz. thanx

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This is not much of a «for sure and fast plz thanx» kind of site, by the way. – Mariano Suárez-Alvarez Mar 2 2010 at 18:45

I doubt this question has a definitive answer, but Fibonacci discusses this in Liber quadratorum:

I thought about the origin of all square numbers and discovered that they arose from the regular ascent of odd numbers. For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.

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According to L. E. Dickson's History of the theory of numbers, vol. 2, ch. 1, this goes back to Pythagoras (570-501 BC)

Later: Ah... Browsing that book is always fun. «N. Beguelin made a puerile illogical attempt to prove that every number is a sum of three triangular numbers». Poor guy: Dickson does not treat him with any kindness at all...

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 Does Dickson give a citation for this? I think it should be in Aristotle's discussion of gnomons, but I'd like to know exactly where. Presumably Aristotle attributes the result to Pythagoras? – Marty Mar 3 2010 at 2:18 Dickson cites F Hoefer, Histoire des mathematiques, Paris, ed. 2, 1879, ed. 5, 1902, 96-121; W W R Ball, Math. Gazette, 8, 1915, 5-12; M. Cantor, Geschichte Math., 1, ed. 3, 1907, 160-3, 252. I haven't followed up on any of those. – Gerry Myerson Mar 3 2010 at 2:28

Achemides from what I recall but my citations are buried in old discrete math books.

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For sure it was a greek person - you can prove it with a picture after all!

$X~~=1$

$\begin{matrix}33\newline 13\end{matrix}= 1+3$

$\begin{matrix} 555\newline 335\newline 135\end{matrix}= 1+3+5$

$\begin{matrix} 7777\newline 5557\newline 3357\newline 1357\end{matrix}= 1+3+5+7$

etc.

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The original source, attributing this theorem to the early pythagoreans is Theon of Smyrna. Eduard Hiller: Expositio Rerum Mathematicarum ad legendum Platonem utilium. Rec. Theon Smyrnaeus, reprinted by Teubner, Stuttgart, 1995.

Further Aristoteles writes, with reference to the early pythagoreans too: "the gnomons are placed round the one" explaining in a somewhat dark manner the geometric aspect of the sum of odd numbers, placed around the 1 Physics, book 3, chapter 4 as sketched in the answer by Stefan .

So there is no chance to find an individual name of the first inventor other than Pythagoras himself. But it is not clear what he really did. No documents of his are left. (Nicomachus, Aristotle, and Archimedes definitively lived too late.)

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