Diophantine Equation--6th Powers

The 2-1 equation

$$A^6+B^6=C^6$$ (1)

is a special case of Fermat's Last Theorem with $n=6$, and so has no solution. Ekl (1996) has searched and found no solutions to the 2-2

$$A^6+B^6=C^6+D^6$$ (2)

with sums less than $7.25\times 10^{26}$.

No solutions are known to the 3-1 or 3-2 equations. However, parametric solutions are known for the 3-3 equation

$$A^6+B^6+C^6=D^6+E^6+F^6$$ (3)

(Guy 1994, pp. 140 and 142). Known solutions are

$$3^6+ 19^6 +22^6 = 10^6+ 15^6+ 23^6$$ (4)

$$36^6+ 37^6+ 67^6 = 15^6+ 52^6+ 65^6$$ (5)

$$33^6+ 47^6+ 74^6 = 23^6+ 54^6+ 73^6$$ (6)

$$32^6+ 43^6+ 81^6 = 3^6+ 55^6+ 80^6$$ (7)

$$37^6+ 50^6+ 81^6 = 11^6+ 65^6+ 78^6$$ (8)

$$25^6+ 62^6+138^6 = 82^6+ 92^6+135^6$$ (9)

$$51^6+113^6+136^6 = 40^6+125^6+129^6$$ (10)

$$71^6+ 92^6+147^6 = 1^6+132^6+133^6$$ (11)

$$111^6+121^6+230^6 = 26^6+169^6+225^6$$ (12)

$$75^6+142^6+245^6 = 14^6+163^6+243^6$$ (13)

(Rao 1934, Lander et al. 1967).

No solutions are known to the 4-1 or 4-2 equations. The smallest primitive 4-3 solutions are

$$41^6+58^6+73^6 = 15^6+32^6+65^6+ 70^6$$ (14)

$$61^6+62^6+85^6 = 52^6+56^6+69^6+ 83^6$$ (15)

$$61^6+74^6+85^6 = 26^6+56^6+71^6+ 87^6$$ (16)

$$11^6+88^6+90^6 = 21^6+74^6+78^6+ 92^6$$ (17)

$$26^6+83^6+95^6 = 23^6+24^6+28^6+101^6$$ (18)

(Lander et al. 1967). Moessner (1947) gave three parametric solutions to the 4-4 equation. The smallest 4-4 solution is

$$2^6+2^6+9^6+9^6=3^6+5^6+6^6+10^6$$ (19)

(Rao 1934, Lander et al. 1967). The smallest 4-4-4 solution is

$$1^6+34^6+49^6+111^6=7^6+43^6+69^6+110^6 = 18^6+25^6+77^6+109^6$$ (20)

(Lander et al. 1967).

No $n$-1 solutions are known for $n\leq 6$ (Lander et al. 1967). No solution to the 5-1 equation is known (Guy 1994, p. 140) or the 5-2 equation.

No solutions are known to the 6-1 or 6-2 equations.

The smallest 7-1 solution is

$$74^6+234^6+402^6+474^6+702^6+894^6+1077^6=1141^6$$ (21)

(Lander et al. 1967). The smallest 7-2 solution is

$$18^6+22^6+36^6+58^6+69^6+78^6+78^6=56^6+91^6$$ (22)

(Lander et al. 1967).

The smallest primitive 8-1 solutions are

$$8^6+ 12^6+ 30^6+ 78^6+102^6+138^6+165^6+246^6=251^6$$ (23)

$$48^6+111^6+156^6+186^6+188^6+228^6+240^6+426^6=431^6$$ (24)

$$93^6+ 93^6+195^6+197^6+303^6+303^6+303^6+411^6=440^6$$ (25)

$$219^6+255^6+261^6+267^6+289^6+351^6+351^6+351^6=440^6$$ (26)

$$12^6+ 66^6+138^6+174^6+212^6+288^6+306^6+441^6=455^6$$ (27)

$$12^6+ 48^6+222^6+236^6+333^6+384^6+390^6+426^6=493^6$$ (28)

$$66^6+ 78^6+144^6+228^6+256^6+288^6+435^6+444^6=499^6$$ (29)

$$16^6+ 24^6+ 60^6+156^6+204^6+276^6+330^6+492^6=502^6$$ (30)

$$61^6+ 96^6+156^6+228^6+276^6+318^6+354^6+534^6=547^6$$ (31)

$$170^6+177^6+276^6+312^6+312^6+408^6+450^6+498^6=559^6$$ (32)

$$60^6+102^6+126^6+261^6+270^6+338^6+354^6+570^6=581^6$$ (33)

$$57^6+146^6+150^6+360^6+390^6+402^6+444^6+528^6=583^6$$ (34)

$$33^6+ 72^6+122^6+192^6+204^6+390^6+534^6+534^6=607^6$$ (35)

$$12^6+ 90^6+114^6+114^6+273^6+306^6+492^6+592^6=623^6$$ (36)

(Lander et al. 1967). The smallest 8-2 solution is

$$8^6+10^6+12^6+15^6+24^6+30^6+33^6+36^6=35^6+37^6$$ (37)

(Lander et al. 1967).

The smallest 9-1 solution is

$$1^6+17^6+19^6+22^6+31^6+37^6+37^6+41^6+49^6=54^6$$ (38)

(Lander et al. 1967). The smallest 9-2 solution is

$$1^6+5^6+5^6+7^6+13^6+13^6+13^6+17^6+19^6=6^6+21^6$$ (39)

(Lander et al. 1967).

The smallest 10-1 solution is

$$2^6+4^6+7^6+14^6+16^6+26^6+26^6+30^6+32^6+32^6=39^6$$ (40)

(Lander et al. 1967). The smallest 10-2 solution is

$$1^6+1^6+1^6+4^6+4^6+7^6+9^6+11^6+11^6+11^6=12^6+12^6$$ (41)

(Lander et al. 1967).

The smallest 11-1 solution is

$$2^6+5^6+5^6+5^6+7^6+7^6+9^6+9^6+10^6+14^6+17^6=18^6$$ (42)

(Lander et al. 1967).

There is also at least one 16-1 identity,

$1^6+2^6+4^6+5^6+6^6+7^6+9^6+12^6+13^6+15^6$
$\mathop{+}16^6+18^6+20^6+21^6+22^6+23^6=28^6\quad$	(43)

(Martin 1893). Moessner (1959) gave solutions for 16-1, 18-1, 20-1, and 23-1.

References

Ekl, R. L. ``Equal Sums of Four Seventh Powers.'' Math. Comput. 65, 1755-1756, 1996.

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.; Parkin, T. R.; and Selfridge, J. L. ``A Survey of Equal Sums of Like Powers.'' Math. Comput. 21, 446-459, 1967.

Martin, A. ``On Powers of Numbers Whose Sum is the Same Power of Some Number.'' Quart. J. Math. 26, 225-227, 1893.

Moessner, A. ``On Equal Sums of Like Powers.'' Math. Student 15, 83-88, 1947.

Moessner, A. ``Einige zahlentheoretische Untersuchungen und diophantische Probleme.'' Glasnik Mat.-Fiz. Astron. Drustvo Mat. Fiz. Hrvatske Ser. 2 14, 177-182, 1959.

Rao, S. K. ``On Sums of Sixth Powers.'' J. London Math. Soc. 9, 172-173, 1934.