Talk:Jacobi symbol

Untitled

Isn't rule #6 wrong?

Question

Do all the brackets here represent the Legendre symbol? If so, it should say so!

Johnbibby 18:44, 10 October 2006 (UTC)[reply]

No, only the ones in the definition (and it's stated there). The other brackets represent Jacobi symbols.

BrunoX 00:06, 23 November 2006 (UTC)[reply]

#10?

is it just me or rule #10 is wrong since it should be stated that m,n must be relativly prime? —The preceding unsigned comment was added by 217.132.13.228 (talk • contribs) 06:15, 21 February 2007 (UTC)

It's OK. If m and n share a factor, then both sides of the equation are zero. -- EJ 10:09, 21 February 2007 (UTC)[reply]

rewrite

I rewrote the page, added references, explained in more detail just what Jacobi added to teh calculateon of (a|p) Virginia-American (talk) 21:38, 2 March 2008 (UTC)[reply]

October 28

I just cut huge rambling sections of this page, including all the calculations. My objection is that the whole point of the Jacobi symbol is that it is polynomial time computable, and the calculations shown used factorization. —Preceding unsigned comment added by 18.51.7.209 (talk) 00:47, 28 October 2008 (UTC)[reply]

And twice your mass removal has been reverted. There is no consensus here on this talk page for your removal of a huge portion of the article. Discuss it here before you remove it again, please. Ariel♥Gold 00:55, 28 October 2008 (UTC)[reply]

Ok, I made an account and finished writing up how the Jacobi symbol is calculated. The page actually already had this embedded in the 'motivation' section; it's just difficult to read. At this point, I think my little section supersedes both 'further properties' and 'motivation', but I'll leave it to the wiki consensus....

Kingofaustria —Preceding undated comment was added at 04:25, 28 October 2008 (UTC).[reply]

I would also be interested in seeing areas such as the 'motivation' section being dissolved. Currently, it reads too much like a math lecture, which I think is at odds with WP:NOTTEXTBOOK. It's trying to be more educational, but I think it makes it less readily informative. CountingPine (talk) 09:42, 28 October 2008 (UTC)[reply]

I put in the "motivation" section to show how the Legendre symbol can require factorization to calculate, whereas the Jacobi symbol can be calculated as quickly as a gcd.

In fact, I wanted it to be *less* like a typical math lecture, in that most math lectures would simply define the symbol and prove the formulas without giving any reason for doing so.

Virginia-American (talk) 20:04, 28 October 2008 (UTC)[reply]

The problem is, even if it is a good, "atypical" lecture, it still doesn't read like an encyclopedia article should. While it is good to include information on how it can be calculated easily, it shouldn't be done in that style, and it shouldn't cloud the actual definition of the symbol, as it was. CountingPine (talk) 09:50, 4 November 2008 (UTC)[reply]

I've cleaned it up well enough to remove the "story" tag. (I think!) Virginia-American (talk) 16:24, 5 November 2008 (UTC)[reply]

Rule #6

Rule #6 is incorrect, quadratic reciprocity should be something like..

If both n and m are odd, then (n/m) = (m/n) unless both n and m are congruent to 3 mod 4. In this case, (n/m) = -(m/n). —Preceding unsigned comment added by 97.77.52.94 (talk) 02:48, 9 March 2011 (UTC)[reply]

That is, in fact, exactly what it says (see the piecewise part on the righthand side). They also have to be positive coprime. Dcoetzee 15:22, 23 March 2013 (UTC)[reply]

Removing Claim Made in Image Caption

It said on the image caption that any non-zero residue m mod n must have (m|n)=1. I initially couldn't figure out how to prove this, then I found a counterexample: m=10 is not congruent to 0 mod n=15, and it is congruent to 5^2=25 mod 15, so it is a non-zero residue. However, (10|15)=(10|3)(10|5)=1*0=0, not 1. Thus, I'm removing the claim. Any objections? David815 (talk) 15:29, 22 February 2015 (UTC)[reply]

Table

So 9 has no quadratic residues aside from 0? 2² = 4 doesn't make 4 a quadratic residue? Am I missing something? Quite misleading... 2804:14D:BAA1:3E5:51B2:48F8:7C8A:685A (talk) 00:16, 5 May 2016 (UTC)[reply]

Example of calculations

The worked problem begins with "Given that 9907 is prime," but that is unnecessary. The calculation will work merely given that 9907 is odd and positive. Colin McLarty (talk) 13:20, 14 January 2017 (UTC)[reply]

Concern

I've noticed that there might be a mistake in the formula presented, when I tried to use this formula to calculate (2/9) in a Jocobi Symbol it equals (2/3)^2 = 1, which is clearly incorrect. You can see it by squaring each of the numbers 0 through 8, you should notice that the only possible remainders mod 9 are 0,1,4,7 which implies 2 is not a quadratic residue, however according to formula presented on the site it would be. I hope it gets fixed soon. 62.122.115.147 (talk) 15:12, 12 June 2022 (UTC)[reply]

This is not a valid concern.

From Jacobi symbol being one does not follow that its top number is a quadratic residue, and it is clearly stated in the article. Teaktl17 (talk) 16:34, 24 July 2022 (UTC)[reply]

Why are there two tables?

I'd like to remove the first one. Any problems? James in dc (talk) 02:49, 7 December 2023 (UTC)[reply]

Rewrite

I have rewritten this article the main changes are

removed the table from the beginning
the sections have been reordered (table and history at end)
the relation between Legendre and Jacobi symbols has been expanded
the computer programs have been desk-checked
removed Legendre symbol calculation (but say they are impractical)
removed (broken) link to calculator - I couldn't find a replacement that prints out the steps

Please visit my sandbox User:James in dc/sandbox/Jacobi symbol and comment. I will replace the article in a week or so, after I finish with the notes.

James in dc (talk) 23:13, 18 December 2023 (UTC)[reply]

C++ Source code is wrong for (a < 0)

The C++ source code appears to be incorrect for (a < 0). I wrote up my own implementation, and used the C++ source code from this page to test against. When I found discrepancies, I spot-checked several of them, and my results were always right, and this function's results were wrong.

Here's a concise demonstration of one error case on Compiler Explorer (there are many others): https://godbolt.org/z/TGTbvhj1P Charles R. Hogg III (talk) 20:26, 2 November 2024 (UTC)[reply]

I didn't fix it, because there are multiple possible ways to fix it, and I didn't know which way would be clearest. However, I did add a comment in the code as a stopgap, so at least people won't be misled in the meantime. Charles R. Hogg III (talk) 20:39, 2 November 2024 (UTC)[reply]

I've now fixed the code. It turns out that the `%` operator in C++ is the "remainder" operation, not "mod". This means that for negative LHS, it will produce negative values. It's a simple fix: there was only one such operation affected. Charles R. Hogg III (talk) 17:35, 3 November 2024 (UTC)[reply]