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A057096 Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles.

%I A057096 #33 Feb 16 2025 08:32:43
%S A057096 60,480,780,1620,2040,3840,4200,6240,7500,12180,12960,14760,15540,
%T A057096 16320,20580,21060,30720,33600,40260,43740,49920,55080,60000,65520,
%U A057096 66780,79860,92820,97440,97500,103680,113400,118080,120120,124320,130560,131820,164640
%N A057096 Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles.
%C A057096 It is an open question whether any two distinct Pythagorean Triples can have the same product of their sides.
%C A057096 From _Amiram Eldar_, Nov 22 2020: (Start)
%C A057096 Named after the French writer Antoine de Saint-Exupéry (1900-1944).
%C A057096 The problem of finding two distinct Pythagorean triples with the same product was proposed by Eckert (1984). It is equivalent of finding a nontrivial solution of the Diophantine equation x*y*(x^4-y^4) = z*w*(z^4-w^4) (Bremner and Guy, 1988). (End)
%D A057096 Richard K. Guy, "Triangles with Integer Sides, Medians and Area." D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994.
%D A057096 Antoine de Saint-Exupéry, Problème du Pharaon, Liège : Editions Dynamo, 1957.
%H A057096 T. D. Noe, <a href="/A057096/b057096.txt">Table of n, a(n) for n = 1..10000</a>
%H A057096 Andrew Bremner and Richard K. Guy, <a href="https://www.jstor.org/stable/2323442">A Dozen Difficult Diophantine Dilemmas</a>, The American Mathematical Monthly, Vol. 95, No. 1 (1988), pp. 31-36.
%H A057096 Ernest J. Eckert, Problem 994, Crux Mathematicorum, Vol. 10, No. 10 (1984), p. 318, <a href="https://cms.math.ca/publications/crux/issue/?volume=10&amp;issue=10">entire issue</a>.
%H A057096 Richard K. Guy, Comment to Problem 994, Crux Mathematicorum, Vol. 12, No. 5 (1986), p. 109, <a href="https://cms.math.ca/publications/crux/issue/?volume=12&amp;issue=5">entire issue</a>.
%H A057096 Henry Plane, <a href="https://www.apmep.fr/IMG/pdf/Parallelepipede_Plane.pdf">Calcule-moi un parallélépipède...</a>, AMPEP, PLOT No. 22 (2002), pp. 22-23.
%H A057096 Giovanni Resta, <a href="https://www.numbersaplenty.com/set/Saint-Exupery_number">Saint-Exupery numbers</a>.
%H A057096 Antoine de Saint Exupéry, <a href="https://www.antoinedesaintexupery.com/personne/le-probleme-du-pharaon/">Le Problème du Pharaon</a>, Succession Saint Exupéry - d'Agay, 2018.
%H A057096 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>.
%F A057096 a(n) = 60*A057097(n) = A057098(n)*A057099(n)*A057100(n).
%e A057096 a(1) = 3*4*5 = 60.
%t A057096 k=5000000; lst={}; Do[Do[If[IntegerQ[a=Sqrt[c^2-b^2]], If[a>=b, Break[]]; x=a*b*c; If[x<=k, AppendTo[lst,x]]], {b,c-1,4,-1}], {c,5,400,1}]; Union@lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 05 2009 *)
%Y A057096 Cf. A009004, A009012, A009111, A020886, A057097, A057098, A057099, A057100.
%K A057096 nonn
%O A057096 1,1
%A A057096 _Henry Bottomley_, Aug 01 2000