Talk:Congruent number

message: 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)
replies: text: 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 , and he first defines it as an integer and then later gives a "more general definition"  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)
thread_title:  Must a congruent number be an integer? 

message: Isn't <TT style='color:Green'>if p mod 8 = 3</TT> (and others...) instead of <TT style='color:Green'>if p = 3 mod 8</TT> ?
--Vp loreta (talk) 08:04, 3 October 2009 (UTC)
replies: text: "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)
thread_title:  mod 

message: 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)
thread_title: patterns 

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thread_title:  External links modified 

message: 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)
replies: text: 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)
thread_title:  2016 result by Alexander Smith