# Talk:Square-free integer

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Maybe this should really be Square-free integer? -- Walt Pohl 01:40, 2 Mar 2004 (UTC)

The separable polynomial page does use the term more generally.

Charles Matthews 08:17, 2 Mar 2004 (UTC)

There's now a square-free polynomial page, too. I've changed the separable polynomial page to link there instead of here.

Baccala@freesoft.org 06:28, 23 January 2006 (UTC)

## We have a lot of equivalent characterizations already,

I know, but here's another:The number of divisors of a squarefree integer is a power of two.Rich 06:55, 1 November 2006 (UTC)

How about 8? Its 4=2^2 factors are 1, 2, 4 and 8, but it is not square-free. 128.101.10.146 23:22, 7 June 2007 (UTC)

Indeed. Having a number of divisors that is a power of two is a necessary, but not sufficient, condition for being squarefree. Doctormatt 23:39, 7 June 2007 (UTC)

## Loop quantum gravity section

That section does not make much sense. There is something crucial missing from the formulas, but I suspect that it masks a conceptual misapprehension. Is this saying more than "any integer ${\displaystyle a}$ can be uniquely represented as ${\displaystyle a=n^{2}\zeta ,}$ where ${\displaystyle \zeta }$ is square-free"? What is the mathematical statement there, and what is result of some experimental spetroscopy? Unless someone comes up with a really compelling reason, I would propose to remove (or at least move) this section from the article. Arcfrk 07:32, 10 March 2007 (UTC)

I have moved the whacky section from the main text to here. Arcfrk 22:28, 23 March 2007 (UTC)

Application in Loop Quantum Gravity

In the theory of loop quantum gravity area is an observable operator. As a consequence, the area of a quantum surface is quantized. Abhay Ashtekar and his colleagues in 1996 found that three incident edges of spins j1, j2, and j3 at a trivalent vertex generate the patch of area:

${\displaystyle a=\ell _{P}^{2}{\sqrt {2j_{1}(j_{1}+1)+2j_{2}(j_{2}+1)-j_{3}(j_{3}+1)}},}$ where ${\displaystyle \ell _{P}}$ is the Planck length.

The spectroscopy of a canonically quantized black hole showed that the area eigenvalue formula fits into the following reduced formula

${\displaystyle \forall n\in N,a=\ell _{P}^{2}n{\sqrt {2\zeta }}}$

(subject to the identification of repeated numbers) where ${\displaystyle \zeta }$ is a square-free number and ${\displaystyle \{\zeta \}=}$ the set of all square-free numbers.

This helps to expect that black hole Hawking radiation is concentraited on a few lines whose energy is proportional to the square root of square-free numbers.

References
• There is a proof in the reference for this. As far as I learn, the proof is simple and neat anyway. Any number is decomposed into its prime numbers, each to an odd or even power. The even power comes up to make a square number, the odd factors make up a square-free number. (129.97.58.55 22:31, 26 March 2007 (UTC))

## Square-free test

Is it known something about the complexity of testing if an integer is square-free? Maybe some relation with Primality test. 193.144.198.250 (talk) 11:28, 11 March 2014 (UTC)