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 A057096 Saint-Exupéry numbers: ordered products of the three sides of Pythagorean triangles. 8
 60, 480, 780, 1620, 2040, 3840, 4200, 6240, 7500, 12180, 12960, 14760, 15540, 16320, 20580, 21060, 30720, 33600, 40260, 43740, 49920, 55080, 60000, 65520, 66780, 79860, 92820, 97440, 97500, 103680, 113400, 118080, 120120, 124320, 130560, 131820, 164640 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS It is an open question whether any two distinct Pythagorean Triples can have the same product of their sides. From Amiram Eldar, Nov 22 2020: (Start) Named after the French writer Antoine de Saint-Exupéry (1900-1944). 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) REFERENCES 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. Antoine de Saint-Exupéry, Problème du Pharaon, Liège : Editions Dynamo, 1957. LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 Andrew Bremner and Richard K. Guy, A Dozen Difficult Diophantine Dilemmas, The American Mathematical Monthly, Vol. 95, No. 1 (1988), pp. 31-36. Ernest J. Eckert, Problem 994, Crux Mathematicorum, Vol. 10, No. 10 (1984), p. 318, entire issue. Richard K. Guy, Comment to Problem 994, Crux Mathematicorum, Vol. 12, No. 5 (1986), p. 109, entire issue. Henry Plane, Calcule-moi un parallélépipède..., AMPEP, PLOT No. 22 (2002), pp. 22-23. Giovanni Resta, Saint-Exupery numbers. Antoine de Saint Exupéry, Le Problème du Pharaon, Succession Saint Exupéry - d'Agay, 2018. Eric Weisstein's World of Mathematics, Pythagorean Triple. FORMULA a(n) = 60*A057097(n) = A057098(n)*A057099(n)*A057100(n). EXAMPLE a(1) = 3*4*5 = 60. MATHEMATICA 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 *) CROSSREFS Cf. A009004, A009012, A009111, A020886, A057097, A057098, A057099, A057100. Sequence in context: A223461 A088943 A097387 * A246774 A341597 A334306 Adjacent sequences: A057093 A057094 A057095 * A057097 A057098 A057099 KEYWORD nonn AUTHOR Henry Bottomley, Aug 01 2000 STATUS approved

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Last modified September 22 18:10 EDT 2023. Contains 365531 sequences. (Running on oeis4.)