# Talk:Integer triangle

## Merger of Heronian triangle and Integer triangle

It seems to me that there is no real justification in having both Heronian triangle and Integer triangle as separate articles. the two concepts are essentially the same, and it would be more helpful to have the information on both in the same place. The other could be either replaced by a redirect, or reduced to a stub, simply defining the concept and linking to the one substantial article. I am much inclined to make this change, but will wait a few days to see if anyone has any objections or other useful suggestions. JamesBWatson (talk) 19:49, 22 May 2010 (UTC)

I disagree. If they were merged, Heronian triangles would have to be merged into Integer triangles, since the former is a subset of the latter. But it is a big subset of the latter, one that people would often want to go directly to (because, aside from Pythagorean triangles, Heronian triangles are the most famous example of integer triangles). Just like the article on Pythagorean triangles, the article on Heronian triangles is quite long, because a lot is known about them. So I think Heronian triangles deserve their own separate article. Duoduoduo (talk) 21:18, 22 May 2010 (UTC)
Disagree with merger proposal. The two concepts are not the same. Heronian triangles are a subset of integral triangles, but a subset that is notable in its own right, with enough material to merit a separate article. Keep articles separate. Gandalf61 (talk) 09:22, 24 May 2010 (UTC)

## Integer triangles in general

I like the first paragraph of the article, but it might be nice to add a second paragraph that discusses integer triangles in general, thus motivating the rest of the article (which deals with specialized classes). How about something like,

Any triple of integers can serve as the the side lengths of an integer triangle as long as it satisfies the triangle inequality: the longest side is shorter than the sum of the other two sides. The number of integer triangles (up to congruence) with perimeter p is the integer closest to (p^2)/48 when p is even and to (p+3)^2/48 when p is odd. [reference: Tom Jenkyns and Eric Muller, Triangular Triples from Ceilings to Floors, American Mathematical Monthly 107:7 (August 2000) 634-639.] Thus there is one triangle having p = 3, 5, or 6, none with p = 4, and two with p = 7. These numbers form sequence A005044 in the On-Line Encyclopedia of Integer Sequences [1]. Because there is little more that one can say about such triangles in general, people mainly study classes of integer triangles with additional properties.

[Further comment: the Monthly reference says that the authors got the formula from Honsberger's MATHEMATICAL GEMS III; perhaps Honsberger says where he got the formula -- the book is not available to me.] 24.89.64.118 (talk) 22:50, 22 June 2012 (UTC)Chris Fisher

Good suggestion - it has my vote. You may wish to modify last sentence "Because there is little more that one can say ...." to "Because, so far, there is little more that one can say ...." Frank M Jackson (talk) 09:31, 23 June 2012 (UTC)
I have Mathematical Gems III', and this result is given in Chapter 3 Section 2. Honsberger gives two proofs and attributes the result to George E. Andrews. Both proofs use the fact that the number of partitions of n into exactly 3 positive parts is the nearest integer to n^2/12 (sequence A069905 in the OEIS). I think this deserves its own section in the article, so I have taken the liberty of adding it. Gandalf61 (talk) 10:17, 23 June 2012 (UTC)

## Isosceles Heronian Triangles..

The section in the article states "All isosceles Heronian triangles are given by rational multiples of"..

Could someone please check that this is accurate? It seems according to the linked source that the process is a little more complex than that statement makes it out to be..

Thanks.. — Preceding unsigned comment added by 96.57.251.228 (talk) 18:31, 16 July 2012 (UTC)

Looks good to me. If the general method for generating heron triangles using integers m, n, k is modified by making m = n, then it gives the isosceles form quoted in the article once all sides are divided by m. Frank M Jackson (talk) 14:00, 17 July 2012 (UTC)
What about (16,17,17), with area 120? — Preceding unsigned comment added by 73.22.1.2 (talk) 05:04, 2 September 2016 (UTC)

## Table of content

Because Pythagorean triangles are a subset of Heronian triangles and integer triangles with one angle 60 degs or one angle 120 degs are subsets of integer triangles with one angle whose cosine is rational, I was going to alter the table of content to reflect these subsets. Please let me know if there are any objections to doing this.Frank M Jackson (talk) 17:42, 19 January 2013 (UTC)

## Properties of Heronian triangles

The second sentence "Thus every Heronian triangle has an even number of sides of odd length,..." can be incorrectly interpreted. The triple (6,8,10) is Heronian and has no odd sides (although zero could be defined as even number). Two suggested corrections:

The perimeter of a Heronian triangle is always an even number. Thus every primitive Heronian triangle has an even number of sides of odd length and every primitive Heronian triangle has exactly one even side.

or

The perimeter of a Heronian triangle is always an even number. Thus every Heronian triangle has an odd number of sides of even length and every primitive Heronian triangle has exactly one even side.

in the second suggestion odd number of sides means 1 side or 3 sides.Frank M Jackson (talk) 14:16, 28 August 2013 (UTC)

## Reason for reverting recent edit

An edit today says

*It has been conjectured that there exists an integer triangle with one side length corresponding to every prime number>2 (OEIS A229159)

But using our article's definition of an integer triangle as having integer sides, this is trivially true: for prime p, let the sides be e.g. (p, p, p). I think it's clear in the source that the source is using integer triangle to mean Heronian triangle -- i.e., an integer-sided triangle with integer area. But our section on Properties of Heronian triangles has a stronger, known, property:

• There exist an infinite number of primitive Heronian triangles with one side length equal to a provided that a > 2.

Duoduoduo (talk) 03:05, 26 September 2013 (UTC)

## What's Up with the Pythagorean Triangles with Integer Altitude from the Hypotenuse?

Nice job with the formatting of this section to whoever did it, but it was screwed up. It claimed to provide a formula generating all primitive Pythagorean triangles with the property that the altitude to the hypotenuse be integral, by requiring that m and n be coprime, but this failed miserably, as Euclid's formula was not used! Instead, some other construction of a, b, and c was used, one for which m and n being coprime and m-n being odd does not guarantee primitivity. In fact, there are no such triangles: primitive Pythagorean triangles with integer altitude from the hypotenuse, that is. This is immediately obvious from Euclid's formula, after all. In general, though, a Pythagorean triangle has an integer altitude from the hypotenuse iff the length of the hypotenuse of its corresponding primitive triangle divides the scale factor from that primitive triangle to the triangle itself. Thus, that's why I changed that section, sources notwithstanding. David815 (talk) 03:07, 29 March 2015 (UTC)

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