As a consequence of Matiyasevich's refutation of Hilbert's 10th problem, it can be proved that there does not exists a general algorithm for solving a general quartic Diophantine equation. However, the algorithm for constructing such an unsolvable quartic Diophantine equation can require arbitrarily many variables (Matiyasevich 1993).
As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 19 positive biquadrates (
), that every "sufficiently large" integer is a sum of no more than 16 positive biquadrates (
), and that every integer is a sum of at most 10 signed biquadrates ( 
;
equations) are 353, 651, 2487, 2501, 2829, ... (Sloane's A003294).
The 4.1.2 equation
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(1) |
is a case of Fermat's last theorem with n = 4 and therefore has no solutions. In fact, the equations
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(2) |
also have no solutions in integers (Nagell 1951, pp. 227 and 229). The equation
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has no solutions in integers (Nagell 1951, p. 230). The only number of the form
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(4) |
which is prime is 5 (Baudran 1885, Le Lionnais 1983).
Let the notation 
stand for the equation consisting of a sum of m pth powers being equal to a sum of n pth powers. In 1772, Euler 
proposed that the 4.1.3 equation
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(5) |
had no solutions in integers (Lander et al. 1967). This assertion is known as the Euler quartic conjecture. Ward (1948) showed there were no solutions for 
,
by Lander et al. (1967). However, the Euler quartic conjecture was disproved in 1987 by N. Elkies, who, using a geometric construction, found
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and showed that infinitely many solutions existed (Guy 1994, p. 140). In 1988, Roger Frye found
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and proved that there are no solutions in smaller integers (Guy 1994, p. 140). Another solution was found by Allan MacLeod in 1997,
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(Ekl 1998). It is not known if there is a parametric solution. In contrast, there are many solutions to the equation
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(see below).
The 4.1.4 equation
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has solutions
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(13) |
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(14) |
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(15) |
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(16) |
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(17) |
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(18) |
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(19) |
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(22) |
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(26) |
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(27) |
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(28) |
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(29) |
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(30) |
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(31) |
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(32) |
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(35) |
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(36) |
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(40) |
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(44) |
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(45) |
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(48) |
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(49) |
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(50) |
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(51) |
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(52) |
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(53) |
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(54) |
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(55) |
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(56) |
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(57) |
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(58) |
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(59) |
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(61) |
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(67) |
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(69) |
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(70) |
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(75) |
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(76) |
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(79) |
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(80) |
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(83) |
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(84) |
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(85) |
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(86) |
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(87) |
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(88) |
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(89) |
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(90) |
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(91) |
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(92) |
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(93) |
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(94) |
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(95) |
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(96) |
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(97) |
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(Norrie 1911, Patterson 1942, Leech 1958, Brudno 1964, Lander et al. 1967, Rose and Brudno 1973; A. Stinchcombe, pers. comm., Oct. 25, 2004), but it is not known if there is a parametric solution (Guy 1994, p. 139).
There are an infinite number of solutions to the 4.1.5 equation
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(99) |
Some of the smallest are
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(100) |
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(101) |
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(102) |
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(103) |
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(104) |
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(106) |
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(Berndt 1994). Berndt and Bhargava (1993) and Berndt (1994, pp. 94-96) give Ramanujan's
solutions for arbitrary s, t, m, and n,
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(108) |
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Parametric solutions to the 4.2.2 equation
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(110) |
are known (Euler 1802; Gérardin 1917; Guy 1994, pp. 140-141), but no "general" solution is known (Hardy 1999, p. 21). A few specific solutions are
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(111) |
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(112) |
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(113) |
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(114) |
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(115) |
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(116) |
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(117) |
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(118) |
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(Sloane's A003824 and A018786; Richmond 1920; Dickson, pp. 60-62; Dickson 1966, pp. 644-647; Leech 1957; Berndt 1994, p. 107; Ekl 1998 [with typo]), the smallest of which is due to Euler (Hardy 1999, p. 21). Lander et al. (1967) give a list of 25 primitive 4.2.2 solutions. General (but incomplete) solutions are given by
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(120) |
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(121) |
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(122) |
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where
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(124) |
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(125) |
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(126) |
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(127) |
(Hardy and Wright 1979).
Parametric solutions to the 4.2.3 equation
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(128) |
are known (Gérardin 1910, Ferrari 1913). The smallest solution is
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(129) |
(Lander et al. 1967).
Ramanujan gave the 4.2.4 equation
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(130) |
Ramanujan gave the 4.3.3 equations
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(131) |
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(132) |
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(133) |
(Berndt 1994, p. 101). Similar examples can be found in Martin (1896). Parametric solutions were given by Gérardin (1911).
Ramanujan also gave the general expression
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(134) |
(Berndt 1994, p. 106). Dickson (1966, pp. 653-655) cites several formulas giving solutions to the 4.3.3 equation, and Haldeman (1904) gives a general formula.
Ramanujan gave the 4.3.4 identities
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(135) |
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(136) |
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(137) |
(Berndt 1994, p. 101). Haldeman (1904) gives general formulas for 4-2 and 4-3 equations.
Ramanujan gave
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(138) |
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(139) |
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(140) |
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(141) |
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(142) |
(Berndt 1994, pp. 96-97). Formula (139) is equivalent to Ferrari's identity
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(143) |
Bhargava's theorem is a general identity which gives the above equations as a special case, and may have been the route by which Ramanujan proceeded. Another identity due to Ramanujan is
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(144) |
, V. Kyrtatas (pers. comm., June 19, 1997) noticed that 
satisfy
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(145) |
and asked if there are any other distinct integer solutions. Additional solutions are 
and 
(E. Clark, pers. comm., Jan. 26, 2004) and 
,
,
(A. Stinchcombe, pers. comm., Nov. 19, 2004).
Bhargava's Theorem, Biquadratic Number, Ford's Theorem, Multigrade Equation, Waring's Problem
Barbette, E. Les sommes de p-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 
.
Chen, S. "Equal Sums of Like Powers: On the Integer Solution of the Diophantine System." http://www.nease.net/~chin/eslp/.
Dickson, L. E. Introduction to the Theory of Numbers. New York: Dover.
Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1966.
Dutch, S. "Power Page: Sums of Fourth Powers." http://www.uwgb.edu/dutchs/RECMATH/rmpowers.htm#4power.
Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.
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. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.
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 .
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.
Meyrignac, J.-C. "Computing Minimal Equal Sums Of Like Powers." http://euler.free.fr.
Nagell, T. "Some Diophantine Equations of the Fourth Degree with Three Unknowns" and "The Diophantine Equation 
.
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 
.
Rivera, C. "Problems & Puzzles: Puzzle 047- 
,
.
Rose, K. and Brudno, S. "More About Four Biquadrates Equal One Biquadrate." Math. Comput. 27, 491-494, 1973.
Sloane, N. J. A. Sequences A003294/M5446, A003824, and A018786 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
Ward, M. "Euler's Problem on Sums of Three Fourth Powers." Duke Math. J. 15, 827-837, 1948.