Call an equation involving quartics  if a sum of quartics is equal to a sum of fourth Powers. The 21 equation
(1) 
(2) 
Parametric solutions to the 22 equation
(3) 
(4)  
(5)  
(6)  
(7)  
(8)  
(9)  
(10)  
(11)  
(12) 
(13)  
(14)  
(15)  
(16) 
(17)  
(18)  
(19)  
(20) 
In 1772, Euler proposed that the 31 equation
(21) 
(22) 
(23) 
(24) 
In contrast, there are many solutions to the 31 equation
(25) 
Parametric solutions to the 32 equation
(26) 
(27) 
Ramanujan gave the 33 equations
(28)  
(29)  
(30) 
Ramanujan also gave the general expression
(31) 
The 41 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) 
Ramanujan gave the 42 equation
(56) 
(57)  
(58)  
(59) 
There are an infinite number of solutions to the 51 equation
(60) 
(61)  
(62)  
(63)  
(64)  
(65)  
(66)  
(67)  
(68) 
(69) 

(70) 
Ramanujan gave
(71) 
(72) 
(73) 
(74) 
(75) 
(76) 
(77) 
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) 
See also Bhargava's Theorem, Ford's Theorem
References
Barbette, E. Les sommes de iémes puissances distinctes égales à une piéme puissance. Doctoral Dissertation, Liege, Belgium. Paris: GauthierVillars, 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: SpringerVerlag, 1994.
Berndt, B. C. and Bhargava, S. ``RamanujanFor Lowbrows.'' Am. Math. Monthly 100, 645656, 1993.
Bhargava, S. ``On a Family of Ramanujan's Formulas for Sums of Fourth Powers.'' Ganita 43, 6367, 1992.
Brudno, S. ``A Further Example of .'' Proc. Cambridge Phil. Soc. 60, 10271028, 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 17951796 13, 45, 1802.
Fauquembergue, E. L'intermédiaire des Math. 5, 33, 1898.
Ferrari, F. L'intermédiaire des Math. 20, 105106, 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: SpringerVerlag, pp. 139144 and 192193, 1994.
Haldeman, C. B. ``On Biquadrate Numbers.'' Math. Mag. 2, 285296, 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, 5255, 1998.
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. ``A Survey of Equal Sums of Like Powers.'' Math. Comput. 21, 446459, 1967.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983.
Leech, J. ``Some Solutions of Diophantine Equations.'' Proc. Cambridge Phil. Soc. 53, 778780, 1957.
Leech, J. ``On .'' Proc. Cambridge Phil. Soc. 54, 554555, 1958.
Martin, A. ``About Biquadrate Numbers whose Sum is a Biquadrate.'' Math. Mag. 2, 173184, 1896.
Martin, A. ``About Biquadrate Numbers whose Sum is a BiquadrateII.'' Math. Mag. 2, 325352, 1904.
Norrie, R. University of St. Andrews 500th Anniversary Memorial Volume. Edinburgh, Scotland: pp. 8789, 1911.
Patterson, J. O. ``A Note on the Diophantine Problem of Finding Four Biquadrates whose Sum is a Biquadrate.'' Bull. Amer. Math. Soc. 48, 736737, 1942.
Ramanujan, S. Notebooks. New York: SpringerVerlag, pp. 385386, 1987.
Richmond, H. W. ``On Integers Which Satisfy the Equation .'' Trans. Cambridge Phil. Soc. 22, 389403, 1920.
Sloane, N. J. A. A003824, A018786, and A003294/M5446 in ``An OnLine 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, 827837, 1948.
© 19969 Eric W. Weisstein