| 释义 |
Diophantine Equation--CubicThe 2-1 equation
 | (1) |
 . In fact, this particular case was known not to have any solutionslong before the general validity of Fermat's Last Theorem was established. The 2-2 equation
 | (2) |
 ,
 |  |  | (3) |  | |  | (4) |  | |  | (5) |  | |  | (6) |  | |  | (7) |  | |  | (8) |  | |  | (9) |  | |  | (10) |  | |  | (11) |  | |  | (12) |
(Sloane's A001235; Moreau 1898). The first number (Madachy 1979, pp. 124 and 141) in this sequence, the so-calledRamanujan  byG. H. Hardy, but was known as early as 1657 (Berndt and Bhargava 1993). The smallest numberrepresentable in  ways as a sum of cubes is called the th Taxicab Number.
Ramanujan gave a general solution to the 2-2 equation as
 | (13) |
 | (14) |
 | |  | (15) |
Hardy and Wright (1979, Theorem 412) prove that there are numbers that are expressible as the sum of two cubes in ways for any (Guy 1994, pp. 140-141). The proof is constructive, providing a method for computing such numbers:given Rationals Numbers  and  , compute
Then
 | (20) |
 is picked to belarge enough, the  and  will be Positive. If is still larger, the  will be large enough for and to be used as the inputs to produce a third pair, etc. However, the resulting integers may be quite large, even for . E.g., starting with  , the algorithm finds
 | (21) |
The numbers representable in three ways as a sum of two cubes (a 2-2-2 equation) are
(Guy 1994, Sloane's A003825). Wilson (1997) found 32 numbers representable in four ways as the sum of two cubes (a2-2-2-2 equation). The first is | |  | (29) |
The smallest known numbers so representable are 6963472309248, 12625136269928, 21131226514944, 26059452841000,... (Sloane's A003826). Wilson also found six five-way sums,and a single six-way sum
The first rational solution to the 3-1 equation
 | (37) |
This is equivalent to the general 2-2 solution found by Ramanujan (Berndt 1994, pp. 54 and 107). The smallest integralsolutions are
 | |  | (40) |  | |  | (41) |  | |  | (42) |  | |  | (43) |  | |  | (44) |  | |  | (45) |  | |  | (46) |
(Beeler et al. 1972; Madachy 1979, pp. 124 and 141). Other general solutions have been found by Binet (1841) and Schwering(1902), although Ramanujan's formulation is the simplest. No general solution giving allPositive integral solutions is known (Dickson 1966, pp. 550-561).
4-1 equations include
A solution to the 4-4 equation is
 | (49) |
5-1 equations
and a 6-1 equation is given by
 | (52) |
 | (53) |
Euler gave the general solution to
 | (54) |
See also Cannonball Problem, Hardy-Ramanujan Number,Super-3 Number, Taxicab Number, Trimorphic Number
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 58 in HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994. Berndt, B. C. and Bhargava, S. ``Ramanujan--For Lowbrows.'' Amer. Math. Monthly 100, 645-656, 1993. Binet, J. P. M. ``Note sur une question relative à la théorie des nombres.'' C. R. Acad. Sci. (Paris) 12, 248-250, 1841. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1966. Guy, R. K. ``Sums of Like Powers. Euler's Conjecture.'' §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-144, 1994. Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 68, 1959. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, 1979. Moreau, C. ``Plus petit nombre égal à la somme de deux cubes de deux façons.'' L'Intermediaire Math. 5, 66, 1898. Schwering, K. ``Vereinfachte Lösungen des Eulerschen Aufgabe:  .'' Arch. Math. Phys. 2, 280-284, 1902. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 157, 1993. Sloane, N. J. A.A001235 andA003825 in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html. Wilson, D. Personal communication, Apr. 17, 1997.
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