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%I #14 Apr 19 2022 01:38:53
%S 13123110,2203385574390,2570042985510,8943387723270,826290896699730,
%T 9381843970167926138271390
%N Areas of triples of primitive Pythagorean triangles having the same area.
%C The generators for the currently known triples are:
%C a(1): (5,138),(38,77),(55,78): Charles Shedd, 1945
%C a(2): (55,3422),(61,3306),(266,2035): _Randall L Rathbun_, 1986
%C a(3): (143,2622),(869,1610),(1817,2002): _Randall L Rathbun_, 1986
%C a(4): (198,3565),(1166,2201),(2035,2438): _Randall L Rathbun_, 1986
%C a(5): (731,10434),(1122,9077),(2465,7238): _Dan Hoey_, May 18 1990
%C a(6): (352538,2999447),(1931103,2398838),(3063347,3215070): Duncan Moore, Mar 01 2017
%C The generator (a,b) gives the Pythagorean triangle (b^2+a^2,b^2-a^2,2ab) with area ab(b^2-a^2).
%D R. K. Guy, Unsolved Problems in Number Theory, D21.
%e The generators of a(1) give the 3 Pythagorean triangles (19069,19019,1380), (7373,4485,5852) and (9109,3059,8580). They have the areas 19019*1380/2 = 4485*5852/2 = 3059*8580/2 = 13123110 = a(1).
%Y Cf. A009111.
%K nonn,hard,more
%O 1,1
%A _Duncan Moore_, Mar 04 2017