A248108 Areas of primitive Heronian triangles K which are perfect squares.

36, 900, 7056, 32400, 41616, 44100, 54756, 63504, 69696, 108900, 112896, 176400, 213444, 298116, 396900, 435600, 509796, 608400, 705600, 736164, 756900, 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

Offset: 1

Author

Frank M Jackson, Oct 01 2014

Keywords

• nonn

Comments

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.

Examples

			The first term 36 corresponds to the 5th term of A083875, 6 (area/6).
a(14) = 298116 = 546^2. There are two such Heronian triangles; they have sides (1183,865,696) and (7202,4395,2809).
		

Links

• Will Jagy, Areas of primitive Heron Triangles that are perfect squares, Math Stackexchange, 2014.
• Sascha Kurz, On the Generation of Heronian triangles, University of Bayreuth, Germany, 2008.
• Sascha Kurz, Lists of primitive Heronian triangles, University of Bayreuth, Germany, 2012.
• Jaap Top, Noriko Yui, Congruent number problems and their variants, University of Leiden, Netherlands, 2008.

Crossrefs

Cf. A083875, A247381.