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

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

Offset: 1

Author

Henry Bottomley, Aug 01 2000

Keywords

• nonn

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)

Examples

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

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.

Crossrefs

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

Programs

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 *)

Formula

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