Gas pycnometer - Revision history

en>Neøn: Added /* See also */

2014-08-30T08:54:21Z

Added See also

← Older revision		Revision as of 10:54, 30 August 2014
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	~~In 1834~~, [~~[Carl Gustav Jakob Jacobi~~]~~] found an exact formula~~ for ~~the total number of ways~~ a ~~given positive integer ''n'' can be represented as the sum~~ of ~~four squares~~. ~~Two representations~~ are ~~considered different if their terms~~ are in ~~different order or if the integer being squared (~~not ~~just~~ the ~~square) is different;~~ to ~~illustrate~~, ~~these are three~~ of the ~~eight different ways~~ to ~~represent 1~~:

	~~:<math>~~
	~~\begin{align}~~
	& ~~1^2~~ + ~~0^2~~ + ~~0^2~~ + ~~0^2 \\~~
	& ~~0^2~~ + ~~1^2~~ + ~~0^2~~ + ~~0^2 \\~~
	~~& (~~-~~1)^2~~ + ~~0^2~~ + ~~0^2~~ + ~~0^2~~.
	~~\end{align}~~
	~~</math>~~

	~~The number~~ of ~~ways~~ to ~~represent n as~~ the ~~sum~~ of ~~four squares~~ is ~~eight times~~ the ~~sum~~ of the ~~[[divisor]]s~~ of ~~''n'' if ''n'' is odd~~ and ~~24 times~~ the ~~sum~~ of the ~~odd divisors~~ of ~~''n'' if ''n''~~ is ~~even (see [[divisor function]]), i~~.e.

	~~: <math>r_4(n)=\begin{cases}8\sum\limits_{m\|n}m&\text{if }n\text{~~ is ~~odd}\\[12pt]~~
	~~24\sum\limits_{\begin{smallmatrix} m\|n \\ m\text{ odd} \end{smallmatrix}}m&\text{if }n\text{~~ is even}.
	~~\end{cases}</math>~~

	~~Equivalently, it is eight times~~ the ~~sum~~ of ~~all its divisors which~~ are not ~~divisible by 4, i~~.e.

	~~: <math>r_4(n)~~=~~8\sum_{m\,~~:~~\, 4\nmid m\|n}m~~.</~~math>~~

	~~In particular~~, ~~for a~~ [~~[prime number]] ''p'' we have the explicit formula ''r''<sub>4<~~/~~sub>(''p'')~~&~~nbsp;~~=~~ 8(''p'' + ~~1).

	~~The proof shows that the [[Theta series#Generalizations\|Theta series]~~] for ~~the [[lattice (discrete subgroup)\|lattice]] '''Z'''~~<~~sup~~>4<~~/sup~~> is ~~a [[modular form]]~~ of a certain ~~level~~, and ~~hence equals~~ a ~~[[linear combination]]~~ of ~~[[Eisenstein series]]~~.

	~~== See also ==~~
	*[[Lagrange's four-square theorem]]
	*[[Lambert series]]

	~~==References==~~
	*{{cite journal\|first=Michael D.~~\|last=Hirschhorn\|coauthors=James A~~. ~~Mcgowan\|title=Algebraic consequences~~ of ~~Jacobi’s two–~~ and ~~four–square theorems\|url=http://citeseerx~~.~~ist~~.~~psu~~.~~edu/viewdoc/summary?doi=10~~.1.1.26.~~9028\|journal=Ismail (eds)~~, ~~Developments in Mathematics\|pages=107–132}}~~
	*{{cite journal\|first=Michael D.~~\|last=Hirschhorn\|title=A simple proof of Jacobi’s four~~-~~square theorem\|date=1987\|journal=Proc~~. ~~Amer~~. ~~Math~~. ~~Soc}}~~

	~~==External links ==~~
	*[http://www.~~math~~.~~ohio~~-~~state~~.~~edu/~econrad/Jacobi/sumofsq/sumofsq~~.~~html Eric Conrad's page]~~

	~~[[Category:Theorems in number theory]]~~

en>Mogism: /* Notes */Cleanup/Typo fixing, typo(s) fixed: greek → Greek using AWB

2013-11-30T17:34:57Z

Notes: Cleanup/Typo fixing, typo(s) fixed: greek → Greek using AWB

← Older revision		Revision as of 19:34, 30 November 2013
Line 1:		Line 1:
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			In 1834, [[Carl Gustav Jakob Jacobi]] found an exact formula for the total number of ways a given positive integer ''n'' can be represented as the sum of four squares. Two representations are considered different if their terms are in different order or if the integer being squared (not just the square) is different; to illustrate, these are three of the eight different ways to represent 1:

			:<math>
			\begin{align}
			& 1^2 + 0^2 + 0^2 + 0^2 \\
			& 0^2 + 1^2 + 0^2 + 0^2 \\
			& (-1)^2 + 0^2 + 0^2 + 0^2.
			\end{align}
			</math>

			The number of ways to represent n as the sum of four squares is eight times the sum of the [[divisor]]s of ''n'' if ''n'' is odd and 24 times the sum of the odd divisors of ''n'' if ''n'' is even (see [[divisor function]]), i.e.

			: <math>r_4(n)=\begin{cases}8\sum\limits_{m\|n}m&\text{if }n\text{ is odd}\\[12pt]
			24\sum\limits_{\begin{smallmatrix} m\|n \\ m\text{ odd} \end{smallmatrix}}m&\text{if }n\text{ is even}.
			\end{cases}</math>

			Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.

			: <math>r_4(n)=8\sum_{m\,:\, 4\nmid m\|n}m.</math>

			In particular, for a [[prime number]] ''p'' we have the explicit formula ''r''<sub>4</sub>(''p'') = 8(''p'' + 1).

			The proof shows that the [[Theta series#Generalizations\|Theta series]] for the [[lattice (discrete subgroup)\|lattice]] '''Z'''<sup>4</sup> is a [[modular form]] of a certain level, and hence equals a [[linear combination]] of [[Eisenstein series]].

			== See also ==
			*[[Lagrange's four-square theorem]]
			*[[Lambert series]]

			==References==
			*{{cite journal\|first=Michael D.\|last=Hirschhorn\|coauthors=James A. Mcgowan\|title=Algebraic consequences of Jacobi’s two– and four–square theorems\|url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.9028\|journal=Ismail (eds), Developments in Mathematics\|pages=107–132}}
			*{{cite journal\|first=Michael D.\|last=Hirschhorn\|title=A simple proof of Jacobi’s four-square theorem\|date=1987\|journal=Proc. Amer. Math. Soc}}

			==External links ==
			*[http://www.math.ohio-state.edu/~econrad/Jacobi/sumofsq/sumofsq.html Eric Conrad's page]

			[[Category:Theorems in number theory]]

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