Talk:Lagrange's four-square theorem

From Wikipedia, the free encyclopedia

Mathematics Portal This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating:	Stub Class	Mid Priority	Field: Number theory
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Maybe start class... Geometry guy 21:39, 9 July 2007 (UTC)

Contents 1 Problem with Jacobi's Combinations 2 Question 3 Removal 4 Proof 5 Fermat's proof?

[edit] Problem with Jacobi's Combinations

I can't seem to find information on this elsewhere, but this seems to fail on very simple cases, such as the number 5. Since 5 is odd, and has only itself as a divisor (or do we count the trivial divisor 1?) We should have 8(5) = 40 possible combinations of sums of squares. But there exists only one: 2² + 1² + 2(0²) = 5 Assuming we allow all permutations of order, that only gives us 4!, and this is not unique to this decomposition either. Allowing squares of negative numbers naively seems to be an unnecessary complication, but I still don't see how this works. Explanation someone please? --71.11.128.29 01:36, 4 January 2007 (UTC)

Actually "allowing all permutations of order" gave you 4!/2 = 12, because 2 of the 4 numbers being permuted are identical(0 and 0). But then allowing pluses and minuses on 1 and 2 multiplies the 12 by 4, giving 48 in all. Since 8 times (1 + 5) = 48, Jacobi's theorem is verified for this example. Hope this helps. Thanks for the thoughtful question.Rich 01:50, 4 January 2007 (UTC)

The article should very clearly explain that Jacobi's formula counts integer (as opposed to natural) solutions. To the best of my knowledge, there is no useful formula for the number of the natural solutions, although Emil Grosswald wrote a book on the subject of counting natural solutions for the case of representations of an integer as a sum of k squares. 129.15.11.123 06:56, 7 March 2007 (UTC)

[edit] Question

Why does it matter if the integers are negative?

negative integers can't be represented as the sum of squares. Horndude77 19:02, 16 July 2005 (UTC)

He was referring to "More formally, for every positive integer n there exist non-negative integers x1, x2, x3, x4". Since negatives square always square up to positives I don't see any reason why the should be non-negative. I've edited accordingly. --poorsod^talk 16:53, 23 May 2007 (UTC)

[edit] Removal

I removed the following:

"In 2005, Zhi-Wei Sun proved that any natural number can be represented as the sum of a square, an even square and a triangular number."

This may or may not be true but why does it have to be here? -- Taku 01:11, July 19, 2005 (UTC)

Word kevinbocking 22:18, 3 July 2007 (UTC)

[edit] Proof

The proof appears in ja article. I might translate it to here if I have time. Anyone interested to see it? -- Taku 07:40, 18 July 2007 (UTC)

[edit] Fermat's proof?

The article says (uncited) "An earlier proof by Fermat was never published." Does this proof definitely exist, or is it another of those marginal notes like the one about his last theorem? AndrewWTaylor 17:20, 24 October 2007 (UTC)