25080: A Number Of Interest

The number of OEIS entries for numbers in the region 25000 can be quite few. For example, 25079 has three entries, 25078 has six, 25077 has five and so on. Thus it was surprising to note that 25080 has an impressive 43 entries. It helps that it's highly factorisable: 2 x 2 x 2 × 3 × 5 × 11 × 19. Let's look at one of these entries.

OEIS A102341: areas of 'close-to-equilateral' integer triangles. To quote from OEIS:

A close-to-equilateral integer triangle is defined to be a triangle with integer sides and integer area such that the largest and smallest sides differ in length by unity. The first five close-to-equilateral integer triangles have sides (5, 5, 6), (17, 17, 16), (65, 65, 66), (241, 241, 240) and (901, 901, 902).

The corresponding sequence of areas for these triangles is 12, 120, 1848, 25080, 351780. So the 'close-to-equilateral' integer triangle with sides 241, 241 and 240 has an integer area of 25080. Here is the image that I posted to my Twitter account, taken from WolframAlpha:

Here is further detail from the OEIS entry:

Heron's formula: a triangle with side lengths \( (x,y,z)  \) has area \(A = \sqrt {s*(s-x)*(s-y)*(s-z)} \) where \(s = (x+y+z)/\)2. For this sequence we assume integer side-lengths \(x = y = z \pm 1 \). Then for \(A \) to also be an integer, \( x+y+z \) must be even, so we can assume \( z = 2k \) for some positive integer \( k \). Now \( s = (x+y+z)/2 = 3k \pm 1 \) and \(A = \sqrt{(3*k \pm 1)*k*k*(k \pm 1)} = k*\sqrt{3*k^2 \pm 4*k + 1} \). To determine when this is an integer, set \( 3*k^2 \pm 4*k + 1 = d^2 \). If we multiply both sides by 3, it is easier to complete the square: \((3*k \pm 2)^2 - 1 = 3*d^2 \). Now we are looking for solutions to the Pell equation \( c^2 - 3*d^2 = 1 \) with \( c = 3*k \pm 2 \), for which there are infinitely many solutions: use the upper principal convergents of the continued fraction expansion of \( \sqrt{3} \). 

 Here are links supplied in the OEIS entry:

•   Danny Rorabaugh, Table of n, a(n) for n = 1..875
•   Steven Dutch, Almost 30-60 Triples
•   ProjectEuler, Problem 94: Almost equilateral triangles
•   Eric Weisstein's World of Mathematics, Heronian Triangle and Pell Equation