Talk:Legendre's three-square theorem

Reverse implication

As currently described, the theorem seems to be $p=4^{a}(8b+7)\implies p=x^{2}+y^{2}+z^{2}\$ However, Dietmann and Elsholtz talk of "a classifical theorem of Legendre and Gauß" which is a bi-implication. I wonder if the reverse implication was added by Gauss? QVVERTYVS (hm?) 15:18, 6 October 2013 (UTC)

@Qwertyus: No, we have $p=4^{a}(8b+7)\implies p\neq x^{2}+y^{2}+z^{2}\$. --Jobu0101 (talk) 09:26, 19 January 2015 (UTC)

Ehm, yes, that's what I meant. Sorry. QVVERTYVS (hm?) 09:38, 19 January 2015 (UTC)

Relationship with the four-square-theorem

The section "Relationship to the four-square theorem" appears totally inaccurate to me. The proof of the three-square theorem is much more involved, and goes much deeper than that of Lagrange four-square theorem.

The classical proof of the former (by Dirichlet in 1850) involves the quadratic reciprocity law, Dirichlet theorem on the primes in arithmetic progressions, and the determination of the class of ternary quadratic forms equivalent to the trivial form.

The proof of the latter is easily presented in a two hours master's course.

So "This theorem lead's to an easy proof of Lagrange's four square theorem" is about like saying that "the prime number theorem leads to an easy proof of Euclid's theorem on the infinity of prime numbers".

Moreover, the reference appealed to for this section is an unpublished manuscript, by an author who totalize 1 (one) publication on MathSciNet (and on a very different subject).

Thus I think this section should be deleted altogether. Sapphorain (talk) 10:18, 24 June 2014 (UTC)

While I agree that a shady reference is cause for consideration, your other argument for removal isn't completely valid. The section (which up until yesterday when I moved it was just a paragraph in the top matter) is not the only proof and doesn't claim to be the only proof. It does, however, show how the two theorems are related, correct? The proof is very easy to follow provided you know the three-square theorem (and because it's on the page about the three-square theorem, one should be familiar with it at that point. And by familiar I don't mean fully understand why.). It links you to the four-square article with the classic proof, but the relationship between the two is shown clearly. If you feel like some more needs to be added about how this isn't the classic proof for the four-square theorem, I would understand. But complete removal on the basis that the three-square theorem is difficult I would debate some more. BeaumontTaz (talk) 13:14, 24 June 2014 (UTC)

I have added a history section to the page, as well as a short section listing the tools used in a classical proof.

Again, the proof of the four-square theorem requires only very elementary algebra, and can be presented in a 1 to 2-hours course. On the other hand, it takes me the better part of a one-semester master's course to complete the proof of the three-square theorem.

So it doesn't make any sense to use the three-square theorem in order to prove the four-square theorem. The argument might have some interest as an exercise to test a student's logical abilities, but it has no encyclopedic value.Sapphorain (talk) 08:57, 25 June 2014 (UTC)

David Eppstein opinion is "I don't see why we can't do both: describe this implication, but then say that the four-square theorem is easier to prove directly." I can (reluctantly!) agree with his position, provided the implication is supported by its appearance in a serious (peer-reviewed) publication. Without that I would regard it as original research and non-encyclopedic. Sapphorain (talk) 06:42, 19 July 2014 (UTC)

I would think a published book would be fine; it doesn't have to be peer-reviewed research. I agree that the current source is dubious. —David Eppstein (talk) 06:50, 19 July 2014 (UTC)

Ok. A book published by a reasonably serious publishing company (and not by the author him- or herself…).Sapphorain (talk) 07:07, 19 July 2014 (UTC)

Only if direction

In the article it says that the only if direction can easily be seen by taking modulo 8. But I think this works only for a=0. Then our expression is 7 mod 8. In the other cases we get a 4 mod 8 which is easily seen to be the sum of three squares mod 8. --Jobu0101 (talk) 09:27, 19 January 2015 (UTC)

Trivially equivalent?

The history section reads:

In 1796 Gauss proved his Eureka theorem that every positive integer n is the sum of 3 triangular numbers; this is trivially equivalent to the fact that 8n+3 is a sum of 3 squares.

I would understand this sentence so that if one has a 3-triangular composition of n, we can determine a 3-square composition of 8n+3 and vice versa. For instance for n=17: When we know the unique 3-triangular composition of 17, to wit (1, 6, 10), we can trivially find a 3-square composition of 131; and if we have one of the three 3-square compositions of 131, to wit any of (1, 9, 121), (25, 25, 81), or (1, 49, 81), we can trivially find a 3-triangula composition of 17. I am awfully sorry that this is too trivial for me. I do not see a path to the solution either way.--Lantani (talk) 13:21, 2 August 2015 (UTC) — Preceding unsigned comment added by Lantani (talk • contribs)

It is indeed trivial that n = m(m+1)/2+k(k+1)/2+r(r+1)2 if and only if 8n+3=(2m+1)^2+(2k+1)^2+(2r+1)^2. (a number congruent to 3 mod 8 can be the sum of three squares only if all three are odd). Sapphorain (talk) 14:44, 2 August 2015 (UTC)