Talk:Square-free integer

Square-free → Square-free integer

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

The separable polynomial page does use the term more generally.

Charles Matthews 08:17, 2 March 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)

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 $a$ can be uniquely represented as $a=n^{2}\zeta ,$ where $\zeta$ 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:

$a=\ell _{P}^{2}{\sqrt {2j_{1}(j_{1}+1)+2j_{2}(j_{2}+1)-j_{3}(j_{3}+1)}},$

where $\ell _{P}$ is the Planck length.

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

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

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

• Spectroscopy of a Canonically Quantized Black Hole

— Preceding unsigned comment added by Arcfrk (talk • contribs) 22:28, 23 March 2007 (UTC)

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))

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)

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)

1 as a squarefree integer

I cannot understand: whether today any mathematicians consider squarefree numbers without 1 or not. Is it possible (for contemporary scientists) to define "squarefree number" as "a number that is the product of integer number of different primes"? --Tamtam90 (talk) 22:06, 5 August 2017 (UTC)

Square free numbers may be defined as products of primes that are all different. This definition is equivalent with the one that is given in the first sentence of the article, as 1 is the empty product of primes. Thus, presently, 1 is always defined to be square free. D.Lazard (talk) 08:30, 6 August 2017 (UTC)

Thank you for your answer. I think I'd ask about this subject Mr. John Derbyshire and Mr. Dennis Hejhal. --Tamtam90 (talk) 23:57, 7 August 2017 (UTC)

squarefull and squareful

This business about having two completely different definitions for « squarefull » and « squareful » (according to the number of « l » of the word) is not confirmed by common practice in mathematical publications. A search in MathSciNet with "squarefull integer" or "squarefull integers", anywhere, yields 10 articles, and with "squarefull number" or "squarefull numbers", anywhere, yields 21 articles. The same searches with only one « l » in « squareful » yield 0, resp. 8 articles. In the last output the definition of « Squareful » is the same as that of « squarefull », except in one single article, in which it is indeed defined as « not squarefree » (but for which the reviewer nevertheless writes « squarefull » with two « l » in his review…). Now OEIS is a wiki, and provides zero source for its page ; as for the page of Mathworld, it provides as unique reference 4 sequences from … OEIS (which in passing do not include the one on « squareful numbers », and all of which explicitly concern « non squarefree numbers » , and not « squareful numbers ») .--Sapphorain (talk) 07:01, 24 August 2020 (UTC)

OK, leave it out. Bubba73 ^{You talkin' to me?} 16:46, 24 August 2020 (UTC)

Moved without any discussion

This page has been moved without discussion from square-free integer to squarefree number. The new title is confusing, as "square-free" is nonsensical when applied to numbers that are not integers. So, I'll ask to revert this move. D.Lazard (talk) 20:36, 24 September 2021 (UTC)

I *think* I've always seen it called "number". Bubba73 ^{You talkin' to me?} 01:52, 25 September 2021 (UTC)

Broken redirect

Cube-free integer redirects to a section of this page that doesn’t exist?? 117.198.96.34 (talk) 07:01, 5 February 2024 (UTC)

You are right. I shall nominate the page for deletion. JBW (talk) 15:20, 5 February 2024 (UTC)

I have now listed this at Wikipedia:Redirects_for_discussion/Log/2024_February_5#Cube free integer. You may contribute to the discussion if you wish to. JBW (talk) 15:37, 5 February 2024 (UTC)

Okay, starting this by saying I am the original topic creator at a different location..

Could we add a section for (x^n-free integers for n higher than 2)?

Would make the redirect make more sense in my opinion. 122.171.18.2 (talk) 11:34, 7 February 2024 (UTC)

Very confusing section

I find the section Square-free factors of integers to be very confusing.

One reason for this is too many different names for three integers.

Another reason for this is that the section begins by asserting it will use the notation q_i as it was defined in the previous section. But it is far from clear what this notation means in the previous section; it is used in this section as if each q_i is a prime number.

I hope someone knowledgeable about this subject can rewrite this section much more clearly. ~2025-32175-13 (talk) 17:14, 12 November 2025 (UTC)

Both sections look perfectly clear and well-written to me, with clearly stated definitions, and I don't see any possible confusion. Maybe the example (in which each q_i is indeed unfortunately a prime number) was not well chosen. For another example, consider $n=37,837,800=2^{3}\cdot 3^{3}\cdot 5^{2}\cdot 7^{2}\cdot 11\cdot 13$. Then one has $q_{1}=143,\;q_{2}=35,\;q_{3}=6$. The square-free part is 143, the square-free factor such that the quotient is a square is 2 ⋅ 3 ⋅ 11 ⋅ 13 = 858, and the largest square-free factor is 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13= 30,030.--Sapphorain (talk) 18:51, 12 November 2025 (UTC)

Digression about polynomials

I removed the following from Square-free_integer#Square-free_factors_of_integers:

• This is a major difference between the arithmetic of the integers, and the arithmetic of the univariate polynomials, as polynomial-time algorithms are known for square-free factorization of polynomials (in short, the largest square-free factor of a polynomial is its quotient by the greatest common divisor of the polynomial and its formal derivative).

The reason is that it's a rather detailed discussion of polynomial factorization, whose context is a discussion of the difference between primality testing vs. factorization of integers. This content belongs in a discussion of polynomials, not integers. Also, there is already an appropriate comparison for the main topic, square-free factorization of polynomials vs. integers, in Square-free_integer#Square-free_factorization. Zaslav (talk) 18:09, 10 May 2026 (UTC)