Diophantine Equation--5th Powers

The 2-1 fifth-order Diophantine equation

$\begin{displaymath} A^5+B^5=C^5 \end{displaymath}$

(1)

is a special case of Fermat's Last Theorem with

, and so has no solution. No solutions to the 2-2 equation

$\begin{displaymath} A^5+B^5=C^5+D^5 \end{displaymath}$

(2)

are known, despite the fact that sums up to $1.02\times 10^{26}$ have been checked (Guy 1994, p. 140), improving on the results on Lander et al. (1967), who checked up to $2.8\times 10^{14}$ . (In fact, no solutions are known for Powers of 6 or 7 either.)

No solutions to the 3-1 equation

$\begin{displaymath} A^5+B^5+C^5=D^5 \end{displaymath}$

(3)

are known (Lander et al. 1967), nor are any 3-2 solutions up to $8\times 10^{12}$ (Lander et al. 1967).

Parametric solutions are known for the 3-3 (Guy 1994, pp. 140 and 142). Swinnerton-Dyer (1952) gave two parametric solutions to the 3-3 equation but, forty years later, W. Gosper discovered that the second scheme has an unfixable bug. The smallest primitive 3-3 solutions are

$\displaystyle 24^5+28^5+ 67^5$	$\textstyle =$	$\displaystyle 3^5+54^5+ 62^5$	(4)
$\displaystyle 18^5+44^5+ 66^5$	$\textstyle =$	$\displaystyle 13^5+51^5+ 64^5$	(5)
$\displaystyle 21^5+43^5+ 74^5$	$\textstyle =$	$\displaystyle 8^5+62^5+ 68^5$	(6)
$\displaystyle 56^5+67^5+ 83^5$	$\textstyle =$	$\displaystyle 53^5+72^5+ 81^5$	(7)
$\displaystyle 49^5+75^5+107^5$	$\textstyle =$	$\displaystyle 39^5+92^5+100^5$	(8)

(Moessner 1939, Moessner 1948, Lander et al. 1967).

For 4 fifth Powers, we have the 4-1 equation

$\begin{displaymath} 27^5+84^5+110^5+133^5=144^5 \end{displaymath}$

(9)

(Lander and Parkin 1967, Lander et al. 1967), but it is not known if there is a parametric solution (Guy 1994, p. 140). Sastry's (1934) 5-1 solution gives some 4-2 solutions. The smallest primitive 4-2 solutions are

$\displaystyle 4^5+10^5+20^5+ 28^5$	$\textstyle =$	$\displaystyle 3^5+ 29^5$	(10)
$\displaystyle 5^5+13^5+25^5+ 37^5$	$\textstyle =$	$\displaystyle 12^5+ 38^5$	(11)
$\displaystyle 26^5+29^5+35^5+ 50^5$	$\textstyle =$	$\displaystyle 28^5+ 52^5$	(12)
$\displaystyle 5^5+25^5+62^5+ 63^5$	$\textstyle =$	$\displaystyle 61^5+ 64^5$	(13)
$\displaystyle 6^5+50^5+53^5+ 82^5$	$\textstyle =$	$\displaystyle 16^5+ 85^5$	(14)
$\displaystyle 56^5+63^5+72^5+ 86^5$	$\textstyle =$	$\displaystyle 31^5+ 96^5$	(15)
$\displaystyle 44^5+58^5+67^5+ 94^5$	$\textstyle =$	$\displaystyle 14^5+ 99^5$	(16)
$\displaystyle 11^5+13^5+37^5+ 99^5$	$\textstyle =$	$\displaystyle 63^5+ 97^5$	(17)
$\displaystyle 48^5+57^5+76^5+100^5$	$\textstyle =$	$\displaystyle 25^5+106^5$	(18)
$\displaystyle 58^5+76^5+79^5+102^5$	$\textstyle =$	$\displaystyle 54^5+111^5$	(19)

(Rao 1934, Moessner 1948, Lander et al. 1967).

A two-parameter solution to the 4-3 equation was given by Xeroudakes and Moessner (1958). Gloden (1949) also gave a parametric solution. The smallest solution is

$\begin{displaymath} 1^5+8^5+14^5+27^5=3^5+22^5+25^5 \end{displaymath}$

(20)

(Rao 1934, Lander et al. 1967). Several parametric solutions to the 4-4 equation were found by Xeroudakes and Moessner (1958). The smallest 4-4 solution is

$\begin{displaymath} 5^5+6^5+6^5+8^5=4^5+7^5+7^5+7^5 \end{displaymath}$

(21)

(Rao 1934, Lander et al. 1967). The first 4-4-4 equation is

$\begin{displaymath} 3^5+48^5+52^5+61^5=13^5+36^5+51^5+64^5=18^5+36^5+44^5+66^5 \end{displaymath}$

(22)

(Lander et al. 1967).

Sastry (1934) found a 2-parameter solution for 5-1 equations

$+(10u^3v^2)^5+(50uv^4)^5=(u^5+75v^5)^5\quad$ (23)

(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

$\displaystyle 19^5+ 43^5+ 46^5+ 47^5+ 67^5$	$\textstyle =$	$\displaystyle 72^5$	(24)
$\displaystyle 21^5+ 23^5+ 37^5+ 79^5+ 84^5$	$\textstyle =$	$\displaystyle 94^5$	(25)
$\displaystyle 7^5+ 43^5+ 57^5+ 80^5+100^5$	$\textstyle =$	$\displaystyle 107^5$	(26)
$\displaystyle 78^5+120^5+191^5+259^5+347^5$	$\textstyle =$	$\displaystyle 365^5$	(27)
$\displaystyle 79^5+202^5+258^5+261^5+395^5$	$\textstyle =$	$\displaystyle 415^5$	(28)
$\displaystyle 4^5+ 26^5+139^5+296^5+412^5$	$\textstyle =$	$\displaystyle 427^5$	(29)
$\displaystyle 31^5+105^5+139^5+314^5+416^5$	$\textstyle =$	$\displaystyle 435^5$	(30)
$\displaystyle 54^5+ 91^5+101^5+404^5+430^5$	$\textstyle =$	$\displaystyle 480^5$	(31)
$\displaystyle 19^5+201^5+347^5+388^5+448^5$	$\textstyle =$	$\displaystyle 503^5$	(32)
$\displaystyle 159^5+172^5+200^5+356^5+513^5$	$\textstyle =$	$\displaystyle 530^5$	(33)
$\displaystyle 218^5+276^5+385^5+409^5+495^5$	$\textstyle =$	$\displaystyle 553^5$	(34)
$\displaystyle 2^5+298^5+351^5+474^5+500^5$	$\textstyle =$	$\displaystyle 575^5$	(35)

(Lander and Parkin 1967, Lander et al. 1967).

The smallest primitive 5-2 solutions are

$\displaystyle 4^5+ 5^5+ 7^5+16^5+21^5$	$\textstyle =$	$\displaystyle 1^5+22^5$	(36)
$\displaystyle 9^5+11^5+14^5+18^5+30^5$	$\textstyle =$	$\displaystyle 23^5+29^5$	(37)
$\displaystyle 10^5+14^5+26^5+31^5+33^5$	$\textstyle =$	$\displaystyle 16^5+38^5$	(38)
$\displaystyle 4^5+22^5+29^5+35^5+36^5$	$\textstyle =$	$\displaystyle 24^5+42^5$	(39)
$\displaystyle 8^5+15^5+17^5+19^5+45^5$	$\textstyle =$	$\displaystyle 30^5+44^5$	(40)
$\displaystyle 5^5+ 6^5+26^5+27^5+44^5$	$\textstyle =$	$\displaystyle 36^5+42^5$	(41)

(Rao 1934, Lander et al. 1967).

The 6-1 equation has solutions

$\displaystyle 4^5+ 5^5+ 6^5+ 7^5+ 9^5+11^5$	$\textstyle =$	$\displaystyle 12^5$	(42)
$\displaystyle 5^5+10^5+11^5+16^5+19^5+29^5$	$\textstyle =$	$\displaystyle 30^5$	(43)
$\displaystyle 15^5+16^5+17^5+22^5+24^5+28^5$	$\textstyle =$	$\displaystyle 32^5$	(44)
$\displaystyle 13^5+18^5+23^5+31^5+36^5+66^5$	$\textstyle =$	$\displaystyle 67^5$	(45)
$\displaystyle 7^5+20^5+29^5+31^5+34^5+66^5$	$\textstyle =$	$\displaystyle 67^5$	(46)
$\displaystyle 22^5+35^5+48^5+58^5+61^5+64^5$	$\textstyle =$	$\displaystyle 78^5$	(47)
$\displaystyle 4^5+13^5+19^5+20^5+67^5+96^5$	$\textstyle =$	$\displaystyle 99^5$	(48)
$\displaystyle 6^5+17^5+60^5+64^5+73^5+89^5$	$\textstyle =$	$\displaystyle 99^5$	(49)

(Martin 1887, 1888, Lander and Parkin 1967, Lander et al. 1967).

The smallest 7-1 solution is

$\begin{displaymath} 1^5+7^5+8^5+14^5+15^5+18^5+20^5=23^5 \end{displaymath}$

(50)

(Lander et al. 1967).

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, p. 95, 1994.

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. 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 th-Power Numbers Whose Sum is an th 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.

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.

Rao, K. S. ``On Sums of Fifth Powers.'' J. London Math. Soc. 9, 170-171, 1934.

Sastry, S. ``On Sums of Powers.'' J. London Math. Soc. 9, 242-246, 1934.

Swinnerton-Dyer, H. P. F. ``A Solution of .'' 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.