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# Diophantine Equation--4th Powers

As a consequence of Matiyasevich's refutation of Hilbert's 10th problem, it can be proved that there does not exist 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 (; although it is not known if 10 can be reduced to 9). The first few numbers which are a sum of four fourth powers ( equations) are 353, 651, 2487, 2501, 2829, ... (OEIS A003294).

The 4.1.2 equation

 (1)

is a case of Fermat's last theorem with and therefore has no solutions. In fact, the equations

 (2)

also have no solutions in integers (Nagell 1951, pp. 227 and 229). The equation

 (3)

has no solutions in integers (Nagell 1951, p. 230). The only number of the form

 (4)

which is prime is 5 (Baudran 1885, Le Lionnais 1983).

Let the notation stand for the equation consisting of a sum of th powers being equal to a sum of th powers. In 1772, Euler proposed that the 4.1.3 equation

 (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 , which was subsequently improved to by Lander et al. (1967). However, the Euler quartic conjecture was disproved in 1987 by N. Elkies, who, using a geometric construction, found

 (6)

and showed that infinitely many solutions existed (Guy 1994, p. 140). In 1988, Roger Frye found

 (7)

and proved that there are no solutions in smaller integers (Guy 1994, p. 140). Another solution was found by Allan MacLeod in 1997,

 (8)

(Ekl 1998). It is not known if there is a parametric solution. In contrast, there are many solutions to the equation

 (9)

(see below).

The 4.1.4 equation

 (10)

has solutions

 (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) (76) (77) (78) (79) (80) (81) (82) (83) (84) (85) (86) (87) (88) (89) (90) (91) (92) (93) (94) (95) (96) (97) (98)

(Norrie 1911, Patterson 1942, Leech 1958, Brudno 1964, Lander et al. 1967, Rose and Brudno 1973; A. Stinchcombe, pers. comm., Oct. 25, 2004). Additional solution are given by Wroblewski.

It was not known if there was a parametric solution (Guy 1994, p. 139) until Jacobi and Madden (2008) found an infinite number of solutions to the special case

 (99)

Their solution makes extensive use of elliptic curve theory and the special solution (955, 1770, 2634, 5400; 5491) due to Brudno (1964), which satisfies .

There are an infinite number of solutions to the 4.1.5 equation

 (100)

Some of the smallest are

 (101) (102) (103) (104) (105) (106) (107) (108)

(Berndt 1994). Berndt and Bhargava (1993) and Berndt (1994, pp. 94-96) give Ramanujan's solutions for arbitrary , , , and ,

 (109)

and

 (110)

These are also given by Dickson (2005, 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). Similar quadratic form parametrizations given by Ramanujan can be found using the identity

 (111)

where for or 4, though this is just a special case of an even more general identity (Piezas 2005). The situation is then reduced to finding solutions to , where is the sum and difference of a number of fourth powers. As an example, given

 (112)

then

 (113)

Similarly, given the identity and the equation

 (114)

then there are an infinite number of primitive solutions to the 4.1.6 equation.

Parametric solutions to the 4.2.2 equation

 (115)

are known (Euler 1802; Gérardin 1917; Guy 1994, pp. 140-141), but no "general" solution is known (Hardy 1999, p. 21). The first few primitive solutions are

 (116) (117) (118) (119) (120) (121) (122) (123)

(OEIS A003824; Richmond 1920; Dickson 1957, pp. 60-62; Leech 1957; Berndt 1994, p. 107; Ekl 1998 [with typo]; Dickson 2005, pp. 644-647), 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

 (124)

where

 (125) (126) (127) (128)

and

 (129) (130) (131) (132)

(Hardy and Wright 1979).

Parametric solutions to the 4.2.3 equation

 (133)

are known (Gérardin 1910, Ferrari 1913). The smallest solution is

 (134)

(Lander et al. 1967).

Ramanujan gave the 4.2.4 equation

 (135)

Ramanujan gave the 4.3.3 equations

 (136) (137) (138)

(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

 (139)

(Berndt 1994, p. 106). Dickson (2005, 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

 (140) (141) (142)

(Berndt 1994, p. 101). Haldeman (1904) gives general formulas for 4-2 and 4-3 equations.

Ramanujan gave

 (143)
 (144)
 (145)
 (146)

where

 (147)

(Berndt 1994, pp. 96-97). Formula (◇) is equivalent to Ferrari's identity

 (148)

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

 (149)

where , and 4 may also be replaced by 2 (Ramanujan 1987, Hirschhorn 1998).

V. Kyrtatas (pers. comm., June 19, 1997) noticed that satisfy

 (150)

and asked if there are any other distinct integer solutions. Additional solutions are (285, 2964, 3249, 1769, 1952, 3721) and (185, 1184, 1369, 663, 858, 1521) (E. Clark, pers. comm., Jan. 26, 2004) and (5160, 11481, 16641, 3683, 12446, 16129), (7367, 11954, 19321, 2660, 15029, 17689) (14925, 24676, 39601, 7527, 29722, 37249), (7136, 42593, 49729, 2387, 44702, 47089) (A. Stinchcombe, pers. comm., Nov. 19, 2004).

Bhargava's Theorem, Biquadratic Number, Euler Quartic Conjecture, Ford's Theorem, Multigrade Equation, Waring's Problem

Portions of this entry contributed by Tito Piezas III

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