“Euler Brick”的概念、定义、习题解题方法及翻译-数学辞典

单词

Euler Brick

释义

Euler Brick

A Rectangular Parallelepiped (``Brick'') with integral edges and face diagonals given by

			(1)
			(2)
			(3)

The problem is also called the Brick, Diagonals Problem, Perfect Box, Perfect Cuboid,or Rational Cuboid problem.

Euler found the smallest solution, which has sides , , and and faceDiagonals , , and . Kraitchik gave 257cuboids with the Odd edge less than 1 million (Guy 1994, p. 174). F. Helenius has compiled a list of the 5003smallest (measured by the longest edge) Euler bricks. The first few are (240, 117, 44), (275, 252, 240), (693, 480,140), (720, 132, 85), (792, 231, 160), ... (Sloane'sA031173,A031174,and A031175). Parametric solutions for Eulerbricks are also known.

No solution is known in which the oblique Space Diagonal

(4)

is also an Integer. If such a brick exists, the smallest side must be at least1,281,000,000 (R. Rathbun 1996). Such a solution is equivalent to solving the Diophantine Equations

			(5)
			(6)
			(7)
			(8)

A solution with integral Space Diagonal and two out of three face diagonals is

, and

, giving

, and

. A solution giving integral space and facediagonals with only a single nonintegral Edge is

, and

, giving

, and

See also Cuboid, Cyclic Quadrilateral, Diagonal (Polyhedron),Parallelepiped, Pythagorean Quadruple

References

Guy, R. K. ``Is There a Perfect Cuboid? Four Squares whose Sums in Pairs are Square. Four Squares whose Differences are Square.'' §D18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 173-181, 1994.

Helenius, F. First 1000 Primitive Euler Bricks. notebooks/EulerBricks.dat.

Leech, J. ``The Rational Cuboid Revisited.'' Amer. Math. Monthly 84, 518-533, 1977. Erratum in Amer. Math. Monthly 85, 472, 1978.

Sloane, N. J. A. Sequences A031173,A031174, andA031175in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

Rathbun, R. L. Personal communication, 1996.

Spohn, W. G. ``On the Integral Cuboid.'' Amer. Math. Monthly 79, 57-59, 1972.

Spohn, W. G. ``On the Derived Cuboid.'' Canad. Math. Bull. 17, 575-577, 1974.

Wells, D. G. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin, p. 127, 1986.

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