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Diophantine Equation--5th Powers

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The 5.1.2 fifth-order Diophantine equation

A^5=B^5+C^5
(1)

is a special case of Fermat's last theorem with n=5, and so has no solution. improving on the results on Lander et al. (1967), who checked up to 2.8×10^(14). (In fact, no solutions are known for powers of 6 or 7 either.) No solutions to the 5.1.3 equation

A^5+B^5+C^5=D^5
(2)

are known (Lander et al. 1967). For 4 fifth powers, the 5.1.4 equation has solutions

27^5+84^5+110^5+133^5=144^5
(3)
85282^5+28969^5+3183^5+55^5=85359^5
(4)

(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

(75v^5-u^5)^5+(u^5+25v^5)^5+(u^5-25v^5)^5 +(10u^3v^2)^5+(50uv^4)^5=(u^5+75v^5)^5
(5)

(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

19^5+43^5+46^5+47^5+67^5=72^5
(6)
21^5+23^5+37^5+79^5+84^5=94^5
(7)
7^5+43^5+57^5+80^5+100^5=107^5
(8)
78^5+120^5+191^5+259^5+347^5=365^5
(9)
79^5+202^5+258^5+261^5+395^5=415^5
(10)
4^5+26^5+139^5+296^5+412^5=427^5
(11)
31^5+105^5+139^5+314^5+416^5=435^5
(12)
54^5+91^5+101^5+404^5+430^5=480^5
(13)
19^5+201^5+347^5+388^5+448^5=503^5
(14)
159^5+172^5+200^5+356^5+513^5=530^5
(15)
218^5+276^5+385^5+409^5+495^5=553^5
(16)
2^5+298^5+351^5+474^5+500^5=575^5
(17)

(Lander and Parkin 1967, Lander et al. 1967). The 5.1.6 equation has solutions

4^5+5^5+6^5+7^5+9^5+11^5=12^5
(18)
5^5+10^5+11^5+16^5+19^5+29^5=30^5
(19)
15^5+16^5+17^5+22^5+24^5+28^5=32^5
(20)
13^5+18^5+23^5+31^5+36^5+66^5=67^5
(21)
7^5+20^5+29^5+31^5+34^5+66^5=67^5
(22)
22^5+35^5+48^5+58^5+61^5+64^5=78^5
(23)
4^5+13^5+19^5+20^5+67^5+96^5=99^5
(24)
6^5+17^5+60^5+64^5+73^5+89^5=99^5
(25)

(Martin 1887, 1888, Lander and Parkin 1967, Lander et al. 1967). The smallest 5.1.7 solution is

1^5+7^5+8^5+14^5+15^5+18^5+20^5=23^5
(26)

(Lander et al. 1967).

No solutions to the 5.2.2 equation

A^5+B^5=C^5+D^5
(27)

are known, despite the fact that sums up to 1.02×10^(26) have been checked (Guy 1994, p. 140). The smallest 5.2.3 solution is

14132^5+220^5=14068^5+6237^5+5027^5
(28)

(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

4^5+10^5+20^5+28^5=3^5+29^5
(29)
5^5+13^5+25^5+37^5=12^5+38^5
(30)
26^5+29^5+35^5+50^5=28^5+52^5
(31)
5^5+25^5+62^5+63^5=61^5+64^5
(32)
6^5+50^5+53^5+82^5=16^5+85^5
(33)
56^5+63^5+72^5+86^5=31^5+96^5
(34)
44^5+58^5+67^5+94^5=14^5+99^5
(35)
11^5+13^5+37^5+99^5=63^5+97^5
(36)
48^5+57^5+76^5+100^5=25^5+106^5
(37)
58^5+76^5+79^5+102^5=54^5+111^5
(38)

(Rao 1934, Moessner 1948, Lander et al. 1967). The smallest primitive 5.2.5 solutions are

4^5+5^5+7^5+16^5+21^5=1^5+22^5
(39)
9^5+11^5+14^5+18^5+30^5=23^5+29^5
(40)
10^5+14^5+26^5+31^5+33^5=16^5+38^5
(41)
4^5+22^5+29^5+35^5+36^5=24^5+42^5
(42)
8^5+15^5+17^5+19^5+45^5=30^5+44^5
(43)
5^5+6^5+26^5+27^5+44^5=36^5+42^5
(44)

(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

ax^5+by^5+cz^5=au^5+bv^5+cw^5
(45)

with a+b+c=0. The smallest primitive solutions to the 5.3.3 equation with unit coefficients are

24^5+28^5+67^5=3^5+54^5+62^5
(46)
18^5+44^5+66^5=13^5+51^5+64^5
(47)
21^5+43^5+74^5=8^5+62^5+68^5
(48)
56^5+67^5+83^5=53^5+72^5+81^5
(49)
49^5+75^5+107^5=39^5+92^5+100^5
(50)

(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

1^5+8^5+14^5+27^5=3^5+22^5+25^5
(51)

(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

5^5+6^5+6^5+8^5=4^5+7^5+7^5+7^5
(52)

(Rao 1934, Lander et al. 1967). The first 5.4.4.4 equation is

3^5+48^5+52^5+61^5=13^5+36^5+51^5+64^5
(53)
=18^5+36^5+44^5+66^5
(54)

(Lander et al. 1967).

Moessner and Gloden (1944) give the 5.5.6 solution

22^5+17^5+16^5+6^5+5^5=21^5+20^5+12^5+10^5+2^5+1^5.
(55)

Chen Shuwen found the 5.6.6 solution

87^5+233^5+264^5+396^5+496^5+540^5=90^5+206^5+309^5+366^5+522^5+523^5.
(56)

See also

Multigrade Equation

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References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, p. 95, 1994.Bremner, A. "A Geometric Approach to Equal Sums of Fifth Powers." J. Number Th. 13, 337-354, 1981.Choudhry, A. "The Diophantine Equation ax^5+by^5+cz^5=au^5+bv^5+cw^5." Rocky Mtn. J. Math. 29, 459-462, 1999.Dutch, S. "Sums of Fifth and Higher Powers." http://www.uwgb.edu/dutchs/RECMATH/rmpowers.htm#5power.Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.Gloden, A. "Über mehrgeradige Gleichungen." Arch. Math. 1, 482-483, 1949.Guy, R. K. "Sums of Like Powers. Euler's Conjecture." §D1 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-144, 1994.Lander, L. J. "Geometric Aspects of Diophantine Equations Involving Equal Sums of Like Power." Amer. Math. Monthly 75, 1061-1073, 1968.Lander, L. J. and Parkin, T. R. "A Counterexample to Euler's Sum of Powers Conjecture." Math. Comput. 21, 101-103, 1967.Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446-459, 1967.Martin, A. "Methods of Finding nth-Power Numbers Whose Sum is an nth Power; With Examples." Bull. Philos. Soc. Washington 10, 107-110, 1887.Martin, A. Smithsonian Misc. Coll. 33, 1888.Martin, A. "About Fifth-Power Numbers whose Sum is a Fifth Power." Math. Mag. 2, 201-208, 1896.Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." http://euler.free.fr.Moessner, A. "Einige numerische Identitäten." Proc. Indian Acad. Sci. Sect. A 10, 296-306, 1939.Moessner, A. "Alcune richerche di teoria dei numeri e problemi diofantei." Bol. Soc. Mat. Mexicana 2, 36-39, 1948.Moessner, A. "Due Sistemi Diofantei." Boll. Un. Mat. Ital. 6, 117-118, 1951.Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. École Polytech. de Timisoara 11, 196-219, 1944.Rao, K. S. "On Sums of Fifth Powers." J. London Math. Soc. 9, 170-171, 1934.Sastry, S. and Chowla, S. "On Sums of Powers." J. London Math. Soc. 9, 242-246, 1934.Swinnerton-Dyer, H. P. F. "A Solution of A^5+B^5+C^5=D^5+E^5+F^5." Proc. Cambridge Phil. Soc. 48, 516-518, 1952.Xeroudakes, G. and Moessner, A. "On Equal Sums of Like Powers." Proc. Indian Acad. Sci. Sect. A 48, 245-255, 1958.

Cite this as:

Weisstein, Eric W. "Diophantine Equation--5th Powers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantineEquation5thPowers.html

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