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Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles.
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%I #32 Mar 02 2023 11:57:55

%S 60,480,780,1620,2040,3840,4200,6240,7500,12180,12960,14760,15540,

%T 16320,20580,21060,30720,33600,40260,43740,49920,55080,60000,65520,

%U 66780,79860,92820,97440,97500,103680,113400,118080,120120,124320,130560,131820,164640

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

%C It is an open question whether any two distinct Pythagorean Triples can have the same product of their sides.

%C From _Amiram Eldar_, Nov 22 2020: (Start)

%C Named after the French writer Antoine de Saint-Exupéry (1900-1944).

%C 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 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 Antoine de Saint-Exupéry, Problème du Pharaon, Liège : Editions Dynamo, 1957.

%H T. D. Noe, <a href="/A057096/b057096.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 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 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 Giovanni Resta, <a href="https://www.numbersaplenty.com/set/Saint-Exupery_number">Saint-Exupery numbers</a>.

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>.

%F a(n) = 60*A057097(n) = A057098(n)*A057099(n)*A057100(n).

%e a(1) = 3*4*5 = 60.

%t 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 Cf. A009004, A009012, A009111, A020886, A057097, A057098, A057099, A057100.

%K nonn

%O 1,1

%A _Henry Bottomley_, Aug 01 2000