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Areas of primitive Heronian triangles K which are perfect squares.
0

%I #31 Dec 07 2022 01:51:54

%S 36,900,7056,32400,41616,44100,54756,63504,69696,108900,112896,176400,

%T 213444,298116,396900,435600,509796,608400,705600,736164,756900,

%U 777924,853776,980100,1040400,1192464,1299600,1368900,1920996,2039184,2304324,2340900,2414916,2433600,2683044,2722500,2822400,2944656,3755844,3802500,3920400,4161600,4588164,4769856,5336100,5731236

%N Areas of primitive Heronian triangles K which are perfect squares.

%C It is known that every positive integer is the area of some rational triangle. Hence for every n > 0 there exists at least one primitive Heronian triangle with area K such that n*k^2 = K for some positive integer k. Therefore for the integer 1 there exists primitive Heronian triangles with area K = k^2. This sequence identifies all such occurrences of square areas from lists of primitive Heronian triangles generated by _Sascha Kurz_ (see link). The sequence excludes repetitive terms and is exhaustive as the Kurz lists searched include all primitive Heronian triangles up to a maximum side length of 6000000 and this sequence only includes areas that do not exceed 6000000 (see _T. D. Noe_ comments at A083875). All 46 terms found are displayed. It is conjectured that this sequence is infinite.

%H Will Jagy, <a href="http://math.stackexchange.com/questions/944814">Areas of primitive Heron Triangles that are perfect squares</a>, Math Stackexchange, 2014.

%H Sascha Kurz, <a href="http://www.wm.uni-bayreuth.de/fileadmin/Sascha/Publikationen/On_Heronian_Triangles.pdf">On the Generation of Heronian triangles</a>, University of Bayreuth, Germany, 2008.

%H Sascha Kurz, <a href="http://www.wm.uni-bayreuth.de/index.php?id=554&amp;L=3">Lists of primitive Heronian triangles</a>, University of Bayreuth, Germany, 2012.

%H Jaap Top, Noriko Yui, <a href="http://www.math.leidenuniv.nl/~psh/ANTproc/19yui.pdf">Congruent number problems and their variants</a>, University of Leiden, Netherlands, 2008.

%e The first term 36 corresponds to the 5th term of A083875, 6 (area/6).

%e a(14) = 298116 = 546^2. There are two such Heronian triangles; they have sides (1183,865,696) and (7202,4395,2809).

%Y Cf. A083875, A247381.

%K nonn

%O 1,1

%A _Frank M Jackson_, Oct 01 2014