Talk:Congruent number

Correspondence to $y^{2}=x^{3}-nx$

My memory might not be serving me right, but if I recall correctly there are additional conditions necessary for the x when relating x,y to a,b,c. I think it needs to be a square with an even denominator? Can anyone explain/substantiate this? Crito2161 (talk) 23:44, 30 December 2008 (UTC)

Koblitz uses a different set of relations between (a, b, c) and (x, y) than the ones in the article. Specifically

$a={\sqrt {x+n}}-{\sqrt {x-n}},\quad b={\sqrt {x+n}}+{\sqrt {x-n}},\quad c=2{\sqrt {x}}$

These are connected to the ones in the article by the fact that for a given triangle if P is the point on the elliptic curve obtained by the relations in the article, and P' is the point here (obtained by x = c^2/4), then P' = 2P under the group law. Since now we're looking for doubles of non-trivial rational points on the curve rather than just any non-trivial rational points, some additional criteria have to be met.

Incidently Koblitz specificies the conditions you mentioned that x is a rational square and has even denominator, and "proves" they're sufficient to imply that the point (x, y) generates a triangle by the relations above, but as it turns out he's wrong. Rckrone (talk) 02:28, 22 June 2009 (UTC)

This is fixed in the second edition, where he adds the condition that n and numerator of x are relatively prime. Rckrone (talk) 05:56, 31 July 2009 (UTC)

Must a congruent number be an integer?

The definition at the beginning of the article says a congruent number is an integer, but other parts suggest this may not be necessary.

E.G., the first sentence of the second section says, "The question of determining whether a given rational number is a congruent number is called the Congruent Number Problem." Johnfranks (talk) 19:25, 22 September 2009 (UTC)

The problem considers all positive rationals, but as the intro mentions we only need to determine the result for the equivalence classes Q*/(Q*)², and each of those contains exactly one square-free positive integer. So the search is really limited to square-free positive integers. I'm not sure what the precise definition should be. The article cites the wolfram page, which cites the Koblitz book [1], and he first defines it as an integer and then later gives a "more general definition" [2] that includes positive rationals. So I don't know. I guess the article should mention both then? Rckrone (talk) 20:31, 24 September 2009 (UTC)

mod

Isn't if p mod 8 = 3 (and others...) instead of if p = 3 mod 8 ? --Vp loreta (talk) 08:04, 3 October 2009 (UTC)

"mod" here isn't being used as an operator. The expression is saying that "p = 3" in the context of "mod 8". More precisely: in modulo 8 arithmetic the equivalence class of p is the same as the equivalence class of 3. Rckrone (talk) 02:53, 11 October 2009 (UTC)

patterns

in http://en.wikipedia.org/wiki/Pythagorean_triples there is a nice scatter diagram showing values of a and b for smaller P-triples.

I wonder whether the relationship between primes (mod 8) and congruent numbers have been plotted on that pattern (say in a different colour) and that might lead to an interesting relationship between primes and P-triples David Seed — Preceding unsigned comment added by Davidseed (talk • contribs) 23:07, 30 December 2011 (UTC)

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2016 result by Alexander Smith

Someone should update this page to include Alexander Smith's 2016 result that the congruent numbers have positive density in the natural numbers. He proved that at least 55.9 percent of positive squarefree numbers equivalent to 5, 6, or 7 (mod 8) are congruent numbers:

https://arxiv.org/pdf/1603.08479.pdf

The page appears to be locked, so it seems that an admin has to make the change. — Preceding unsigned comment added by Tjw980 (talk • contribs) 21:53, 10 August 2022 (UTC)

It appears to be an unpublished preprint, still, six years later. That makes it unlikely to pass Wikipedia's standards for what can be used as a reliable source. And in it, it says that it's an undergraduate thesis, so its author is clearly not an established expert in the subject (one of the ways of allowing sources that are not reliably published). —David Eppstein (talk) 22:34, 10 August 2022 (UTC)

That paper is cited by both the American Mathematical Society and the Clay Mathematics Institute:

https://www.ams.org/journals/notices/201911/rnoti-p1875.pdf

https://www.claymath.org/people/alexander-smith

Smith was awarded the 2019 Journal of Number Theory's first-ever David Goss Prize based largely on that work. It's unpublished, but it is entirely verified. — Preceding unsigned comment added by Tjw980 (talk • contribs) 01:31, 11 August 2022 (UTC)

What if a prime p ≡ 1 (mod 8) ?

The last section discusses the three cases of primes p ≡ 3, 5, 7 (mod 8).

But nothing is mentioned about the case of p ≡ 1 (mod 8).

I hope someone knowledgeable about this topic can include some information about primes p ≡ 1 (mod 8).