OFFSET
1,1
COMMENTS
It has been proved that every positive integer is the area of some rational sided Heronian triangle. Therefore for all positive integers n there exists a primitive Heronian triangle such that for some least k^2 its area K = k^2*n. The Mathematica program limits searches to all primitive Heronian triangles whose largest side does not exceed 1000 and returns 0 if no area is found.
LINKS
N. J. Fine, On Rational Triangles, Mathematical Association of America, 83-7 (1976), 517-521.
Jaap Top and Noriko Yui, Congruent number problems and their variants, Algorithmic Number Theory, MSRI Publications Volume 44, 2008, p. 621.
EXAMPLE
a(23)=30^2*23=20700 and the primitive Heronian triangle has sides (73, 579, 598).
MATHEMATICA
getarea[n0_] := (area1=0; Do[If[IntegerQ[area=Sqrt[(a+b+c)(a+b-c)(a-b+c)(-a+b+c)/16]]&&area>0&&IntegerQ[k=Sqrt[area/n0]]&&GCD[a, b, c]==1, area1=area; Break[]], {c, 3, 1000}, {b, 1, c}, {a, 1, b}]; area1); Table[getarea[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Sep 15 2014
EXTENSIONS
Updated and edited by Frank M Jackson, Jun 14 2016
STATUS
approved