The 5.1.2 fifth-order Diophantine equation
(1)
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is a special case of Fermat's last theorem with , and so has no solution. improving on the results on Lander et al. (1967), who checked up to . (In fact, no solutions are known for powers of 6 or 7 either.) No solutions to the 5.1.3 equation
(2)
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are known (Lander et al. 1967). For 4 fifth powers, the 5.1.4 equation has solutions
(3)
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(4)
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(Lander and Parkin 1967, Lander et al. 1967, Ekl 1998), the second of which was found by J. Frye (J.-C. Meyrignac, pers. comm., Sep. 9, 2004), but it is not known if there is a parametric solution (Guy 1994, p. 140). Sastry (1934) found a 2-parameter solution for 5.1.5 equations
(5)
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(quoted in Lander and Parkin 1967), and Lander and Parkin (1967) found the smallest numerical solutions. Lander et al. (1967) give a list of the smallest solutions, the first few being
(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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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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(Lander and Parkin 1967, Lander et al. 1967). The 5.1.6 equation has solutions
(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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(25)
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(Martin 1887, 1888, Lander and Parkin 1967, Lander et al. 1967). The smallest 5.1.7 solution is
(26)
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(Lander et al. 1967).
No solutions to the 5.2.2 equation
(27)
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are known, despite the fact that sums up to have been checked (Guy 1994, p. 140). The smallest 5.2.3 solution is
(28)
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(B. Scher and E. Seidl 1996, Ekl 1998). Sastry's (1934) 5.1.5 solution gives some 5.2.4 solutions. The smallest primitive 5.2.4 solutions are
(29)
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(30)
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(31)
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(32)
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(33)
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(34)
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(35)
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(36)
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(37)
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(38)
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(Rao 1934, Moessner 1948, Lander et al. 1967). The smallest primitive 5.2.5 solutions are
(39)
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(40)
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(41)
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(42)
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(43)
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(44)
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(Rao 1934, Lander et al. 1967).
Parametric solutions are known for the 5.3.3 (Sastry and Lander 1934; Moessner 1951; Swinnerton-Dyer 1952; Lander 1968; Bremmer 1981; Guy 1994, pp. 140 and 142; Choudhry 1999). Swinnerton-Dyer (1952) gave two parametric solutions to the 5.3.3 equation but, forty years later, W. Gosper discovered that the second scheme has an unfixable bug. Choudhry (1999) gave a parametric solution to the more general equation
(45)
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with . The smallest primitive solutions to the 5.3.3 equation with unit coefficients are
(46)
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(47)
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(48)
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(49)
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(50)
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(Moessner 1939, Moessner 1948, Lander et al. 1967, Ekl 1998).
A two-parameter solution to the 5.3.4 equation was given by Xeroudakes and Moessner (1958). Gloden (1949) also gave a parametric solution. The smallest solution is
(51)
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(Rao 1934, Lander et al. 1967).
Several parametric solutions to the 5.4.4 equation were found by Xeroudakes and Moessner (1958).
The smallest 5.4.4 solution is
(52)
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(Rao 1934, Lander et al. 1967). The first 5.4.4.4 equation is
(53)
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(54)
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(Lander et al. 1967).
Moessner and Gloden (1944) give the 5.5.6 solution
(55)
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Chen Shuwen found the 5.6.6 solution
(56)
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