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A256418 Congrua (possible solutions to the congruum problem): numbers k such that there are integers x, y and z with k = x^2-y^2 = z^2-x^2.

%I A256418 #34 Aug 02 2025 18:34:49
%S A256418 24,96,120,216,240,336,384,480,600,720,840,864,960,1080,1176,1320,
%T A256418 1344,1536,1920,1944,2016,2160,2184,2400,2520,2880,2904,3000,3024,
%U A256418 3360,3456,3696,3840,3960,4056,4320,4704,4896,5280,5376,5400,5544
%N A256418 Congrua (possible solutions to the congruum problem): numbers k such that there are integers x, y and z with k = x^2-y^2 = z^2-x^2.
%C A256418 k is a "congruum" iff k/4 is the area of a Pythagorean triangle, so these are the numbers 4*A009112.
%C A256418 Each congruum is a multiple of 24; it cannot be a square.
%C A256418 This entry incorporates many comments that were originally in A057102. A057103 and A055096 need to be checked.
%H A256418 Robert Israel, <a href="/A256418/b256418.txt">Table of n, a(n) for n = 1..10000</a>
%H A256418 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Congruum.html">Congruum</a> (but beware errors)
%H A256418 Wikipedia, <a href="https://en.wikipedia.org/wiki/Congruum">Congruum</a> (but beware errors).
%e A256418 a(11)=840 since 840=29^2-1^2=41^2-29^2 (indeed also 840=37^2-23^2=47^2-37^2).
%t A256418 r[n_] := Reduce[0 < y < x && 0 < x < z && n == x^2 - y^2 == z^2 - x^2, {x, y, z}, Integers];
%t A256418 Reap[For[n = 24, n < 10^4, n += 24, rn = r[n]; If[rn =!= False, Print[n, " ", rn]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Feb 25 2019 *)
%Y A256418 Cf. A004431 for possible values of x in definition. Cf. A057103, A055096 for triangles of all congrua and values of x.
%Y A256418 Cf. A009112, A073120, A135789, A135786.
%K A256418 nonn
%O A256418 1,1
%A A256418 _N. J. A. Sloane_, Apr 06 2015, following a suggestion from _Robert Israel_, Apr 03 2015