The 7.1.2 equation
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(1)
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is a special case of Fermat's last theorem with 
,
and so has no solution. No solutions to the 7.1.3, 7.1.4, 7.1.5, 7.1.6 equations
are known. There is now a known solutions to the 7.1.7 equation,
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(2)
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(M. Dodrill 1999, PowerSum), requiring an update by Guy (1994, p. 140). The smallest 7.1.8 solution is
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(3)
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(Lander et al. 1967, Ekl 1998). The smallest 7.1.9 solution is
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(4)
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(Lander et al. 1967).
No solutions to the 7.2.2, 7.2.3, 7.2.4, or 7.2.5 equations are known. The smallest 7.2.6 equation is
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(5)
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(Meyrignac). The smallest 7.2.8 solution is
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(6)
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(Lander et al. 1967, Ekl 1998). A 7.2.10.10 solution is
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(7)
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(8)
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(Lander et al. 1967).
No solutions to the 7.3.3 equation are known (Ekl 1996), nor are any to 7.3.4. The smallest 7.3.5 equations are
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(9)
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(10)
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No solutions are known to the 7.3.6 equation. The smallest 7.3.7 solution is
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(11)
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(Lander et al. 1967).
Guy (1994, p. 140) asked if a 7.4.4 equation exists. The following solution provide an affirmative answer
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(12)
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(13)
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(14)
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(Ekl 1996, 1998; M. Lau 1999; PowerSum). Numerical solutions to the 7.4.5 equation are given by Gloden (1949). The smallest primitive 7.4.5 solutions are
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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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(20)
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(21)
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(22)
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(Lander et al. 1967, Ekl 1998).
Gloden (1949) gives parametric solutions to the 7.5.5 equation. The first few 7.5.5 solutions are
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(23)
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(24)
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(25)
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(26)
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(27)
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(Lander et al. 1967). Ekl (1998) mentions but does not list 107 primitive solutions to 7.5.5.
A parametric solution to the 7.6.6 equation was given by Sastry and Rai (1948). The smallest is
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(28)
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(Lander et al. 1967). Another found by Chen Shuwen is
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(29)
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Moessner and Gloden (1944) gave the 7.9.10 solution
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(30)
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." §67 in