| 释义 |
Diophantine Equation--QuarticCall an equation involving quartics  - if a sum of quartics is equal to a sum of fourth Powers. The 2-1 equation
 | (1) |
 and therefore has no solutions. In fact, the equations
 | (2) |
Parametric solutions to the 2-2 equation
 | (3) |
 |  |  | (4) |  | |  | (5) |  | |  | (6) |  | |  | (7) |  | |  | (8) |  | |  | (9) |  | |  | (10) |  | |  | | | | | | (11) |  | |  | | | | | | (12) |
(Sloane's A003824and A018786; Richmond 1920, Leech 1957), the smallest of which is due to Euler.  Lander et al. (1967) givea list of 25 primitive 2-2 solutions. General (but incomplete) solutions are given by
where
(Hardy and Wright 1979).
In 1772, Euler proposed that the 3-1 equation
 | (21) |
 , which was subsequently improved to  by Lander et al. (1967). However, the Euler Quartic Conjecture was disproved in 1987 by Noam D. Elkies,who, using a geometric construction, found
 | (22) |
 | (23) |
 | (24) |
In contrast, there are many solutions to the 3-1 equation
 | (25) |
Parametric solutions to the 3-2 equation
 | (26) |
 | (27) |
Ramanujan gave the 3-3 equations
(Berndt 1994, p. 101). Similar examples can be found in Martin (1896). Parametric solutions were given byGérardin (1911).
Ramanujan also gave the general expression
 | (31) |
The 4-1 equation
 | (32) |
 | |  | (33) |  | |  | (34) |  | |  | (35) |  | |  | (36) |  | |  | (37) |  | |  | (38) |  | |  | (39) |  | |  | (40) |  | |  | (41) |  | |  | (42) |  | |  | (43) |  | |  | (44) |  | |  | (45) |  | |  | (46) |  | |  | (47) |  | |  | (48) |  | |  | (49) |  | |  | (50) |  | |  | (51) |  | |  | (52) |  | |  | (53) |  | |  | (54) |  | |  | (55) |
(Norrie 1911, Patterson 1942, Leech 1958, Brudno 1964, Lander et al. 1967), but it is not known if there is a parametricsolution (Guy 1994, p. 139).
Ramanujan gave the 4-2 equation
 | (56) |
(Berndt 1994, p. 101). Haldeman (1904) gives general Formulas for 4-2 and 4-3 equations.
There are an infinite number of solutions to the 5-1 equation
 | (60) |
 | |  | (61) |  | |  | (62) |  | |  | (63) |  | |  | (64) |  | |  | (65) |  | | | (66) |  | |  | (67) |  | |  | (68) |
(Berndt 1994). Berndt and Bhargava (1993) and Berndt (1994, pp. 94-96) give Ramanujan's solutions forarbitrary  ,  , , and ,and | |  | | | (70) |
These are also given by Dickson (1966, p. 649), and two general Formulas are given by Beiler (1966,p. 290). Other solutions are given by Fauquembergue (1898), Haldeman (1904), and Martin (1910).
Ramanujan gave
 | (71) |
 | (72) |
 | (73) |
 | |  | (74) |
where
 | (75) |
 | |  | (76) |
Bhargava's Theorem is a general identity which gives the above equations as a special case, and may have been the routeby which Ramanujan proceeded. Another identity due to Ramanujan is
 | (77) |
 , and 4 may also be replaced by 2 (Ramanujan 1957, Hirschhorn 1998).
V. Kyrtatas noticed that  ,  ,  ,  ,  , and  satisfy
 | (78) |
The first few numbers which are a sum of two or more fourth Powers ( equations) are 353, 651, 2487,2501, 2829, ... (Sloane's A003294). The only number of the form
 | (79) |
References
Barbette, E. Les sommes de  -iémes puissances distinctes égales à une p-iéme puissance. Doctoral Dissertation, Liege, Belgium. Paris: Gauthier-Villars, 1910.Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966. Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994. Berndt, B. C. and Bhargava, S. ``Ramanujan--For Lowbrows.'' Am. Math. Monthly 100, 645-656, 1993. Bhargava, S. ``On a Family of Ramanujan's Formulas for Sums of Fourth Powers.'' Ganita 43, 63-67, 1992. Brudno, S. ``A Further Example of  .'' Proc. Cambridge Phil. Soc. 60, 1027-1028, 1964. Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1966. Euler, L. Nova Acta Acad. Petrop. as annos 1795-1796 13, 45, 1802. Fauquembergue, E. L'intermédiaire des Math. 5, 33, 1898. Ferrari, F. L'intermédiaire des Math. 20, 105-106, 1913. Guy, R. K. ``Sums of Like Powers. Euler's Conjecture'' and ``Some Quartic Equations.'' §D1 and D23 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-144 and 192-193, 1994. Haldeman, C. B. ``On Biquadrate Numbers.'' Math. Mag. 2, 285-296, 1904. Hardy, G. H. and Wright, E. M. §13.7 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979. Hirschhorn, M. D. ``Two or Three Identities of Ramanujan.'' Amer. Math. Monthly 105, 52-55, 1998. Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. ``A Survey of Equal Sums of Like Powers.'' Math. Comput. 21, 446-459, 1967. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. Leech, J. ``Some Solutions of Diophantine Equations.'' Proc. Cambridge Phil. Soc. 53, 778-780, 1957. Leech, J. ``On .'' Proc. Cambridge Phil. Soc. 54, 554-555, 1958. Martin, A. ``About Biquadrate Numbers whose Sum is a Biquadrate.'' Math. Mag. 2, 173-184, 1896. Martin, A. ``About Biquadrate Numbers whose Sum is a Biquadrate--II.'' Math. Mag. 2, 325-352, 1904. Norrie, R. University of St. Andrews 500th Anniversary Memorial Volume. Edinburgh, Scotland: pp. 87-89, 1911. Patterson, J. O. ``A Note on the Diophantine Problem of Finding Four Biquadrates whose Sum is a Biquadrate.'' Bull. Amer. Math. Soc. 48, 736-737, 1942. Ramanujan, S. Notebooks. New York: Springer-Verlag, pp. 385-386, 1987. Richmond, H. W. ``On Integers Which Satisfy the Equation  .'' Trans. Cambridge Phil. Soc. 22, 389-403, 1920. Sloane, N. J. A.A003824,A018786, andA003294/M5446in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html. Ward, M. ``Euler's Problem on Sums of Three Fourth Powers.'' Duke Math. J. 15, 827-837, 1948.
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