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Diophantine Equation--6th Powers

The 6.1.2 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. No 6.1.n solutions are known for n<=6 (Lander et al. 1967; Guy 1994, p. 140). The smallest 6.1.7 solution is

74^6+234^6+402^6+474^6+702^6+894^6+1077^6=1141^6
(2)

(Lander et al. 1967; Ekl 1998). The smallest primitive 6.1.8 solutions are

8^6+12^6+30^6+78^6+102^6+138^6+165^6+246^6 =251^6 48^6+111^6+156^6+186^6+188^6+228^6+240^6+426^6 =431^6 93^6+93^6+195^6+197^6+303^6+303^6+303^6+411^6 =440^6 219^6+255^6+261^6+267^6+289^6+351^6+351^6+351^6 =440^6 12^6+66^6+138^6+174^6+212^6+288^6+306^6+441^6 =455^6 12^6+48^6+222^6+236^6+333^6+384^6+390^6+426^6 =493^6 66^6+78^6+144^6+228^6+256^6+288^6+435^6+444^6 =499^6 16^6+24^6+60^6+156^6+204^6+276^6+330^6+492^6 =502^6 61^6+96^6+156^6+228^6+276^6+318^6+354^6+534^6 =547^6 170^6+177^6+276^6+312^6+312^6+408^6+450^6+498^6 =559^6 60^6+102^6+126^6+261^6+270^6+338^6+354^6+570^6 =581^6 57^6+146^6+150^6+360^6+390^6+402^6+444^6+528^6 =583^6 33^6+72^6+122^6+192^6+204^6+390^6+534^6+534^6 =607^6 12^6+90^6+114^6+114^6+273^6+306^6+492^6+592^6 =623^6
(3)

(Lander et al. 1967). The smallest 6.1.9 solution is

1^6+17^6+19^6+22^6+31^6+37^6+37^6+41^6+49^6=54^6
(4)

(Lander et al. 1967). The smallest 6.1.10 solution is

2^6+4^6+7^6+14^6+16^6+26^6+26^6+30^6+32^6+32^6=39^6
(5)

(Lander et al. 1967). The smallest 6.1.11 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
(6)

(Lander et al. 1967). There is also at least one 6.1.16 identity,

1^6+2^6+4^6+5^6+6^6+7^6+9^6+12^6+13^6+15^6 +16^6+18^6+20^6+21^6+22^6+23^6=28^6
(7)

(Martin 1893). Moessner (1959) gave solutions for 6.1.16, 6.1.18, 6.1.20, and 6.1.23 equations.

Ekl (1996) has searched and found no solutions to the 6.2.2

A^6+B^6=C^6+D^6
(8)

with sums less than 7.25×10^(26). No solutions are known to the 6.2.3 or 6.2.4 equations. The smallest primitive 6.2.5 equations are

1092^6+861^6+602^6+212^6+84^6=1117^6+770^6
(9)
1893^6+1468^6+1407^6+1302^6+1246^6=2041^6+691^6
(10)
2184^6+2096^6+1484^6+1266^6+1239^6=2441^6+752^6
(11)
2653^6+2296^6+1488^6+1281^6+390^6=2827^6+151^6
(12)
2954^6+2481^6+850^6+798^6+420^6=2959^6+2470^6
(13)

(E. Brisse 1999, Resta 1999, Resta and Meyrignac 2003, Meyrignac). The smallest 6.2.6 equation is

241^6+17^6=218^6+210^6+118^6+2·63^6+42^6
(14)

(Ekl 1998). The smallest 6.2.7 solution is

18^6+22^6+36^6+58^6+69^6+78^6+78^6=56^6+91^6
(15)

(Lander et al. 1967). The smallest 6.2.8 solution is

8^6+10^6+12^6+15^6+24^6+30^6+33^6+36^6=35^6+37^6
(16)

(Lander et al. 1967). The smallest 6.2.9 solution is

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

(Lander et al. 1967). The smallest 6.2.10 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
(18)

(Lander et al. 1967).

Parametric solutions are known for the 6.3.3 equation

A^6+B^6+C^6=D^6+E^6+F^6
(19)

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

3^6+19^6+22^6=10^6+15^6+23^6
(20)
36^6+37^6+67^6=15^6+52^6+65^6
(21)
33^6+47^6+74^6=23^6+54^6+73^6
(22)
32^6+43^6+81^6=3^6+55^6+80^6
(23)
37^6+50^6+81^6=11^6+65^6+78^6
(24)
25^6+62^6+138^6=82^6+92^6+135^6
(25)
51^6+113^6+136^6=40^6+125^6+129^6
(26)
71^6+92^6+147^6=1^6+132^6+133^6
(27)
111^6+121^6+230^6=26^6+169^6+225^6
(28)
75^6+142^6+245^6=14^6+163^6+243^6
(29)

(Rao 1934, Lander et al. 1967, Ekl 1998). Ekl (1998) mentions but does not list the 87 smallest solutions to the 6.2.6 equation. The smallest primitive 6.3.4 solutions are

73^6+58^6+41^6=70^6+65^6+32^6+15^6
(30)
85^6+62^6+61^6=83^6+69^6+56^6+52^6
(31)
85^6+74^6+61^6=87^6+71^6+56^6+26^6
(32)
90^6+88^6+11^6=92^6+78^6+74^6+21^6
(33)
95^6+83^6+26^6=101^6+28^6+24^6+23^6
(34)
130^6+44^6+23^6=119^6+108^6+86^6+38^6
(35)
125^6+114^6+38^6=126^6+104^6+93^6+68^6
(36)
205^6+113^6+18^6=198^6+148^6+133^6+39^6
(37)
211^6+123^6+34^6=210^6+134^6+73^6+39^6
(38)
212^6+164^6+103^6=217^6+130^6+114^6+8^6
(39)
222^6+34^6+25^6=217^6+156^6+96^6+68^6
(40)
218^6+167^6+29^6=224^6+107^6+102^6+65^6
(41)
226^6+110^6+17^6=224^6+143^6+72^6+34^6
(42)
244^6+123^6+112^6=238^6+180^6+91^6+72^6
(43)
241^6+172^6+156^6=246^6+145^6+132^6+56^6
(44)
257^6+155^6+6^6=252^6+181^6+143^6+114^6
(45)
265^6+147^6+12^6=231^6+221^6+210^6+114^6
(46)
260^6+218^6+185^6=276^6+152^6+112^6+25^6
(47)
305^6+85^6+66^6=273^6+267^6+172^6+122^6
(48)
312^6+241^6+33^6=315^6+228^6+99^6+2^6
(49)
331^6+234^6+59^6=306^6+294^6+151^6+95^6
(50)
332^6+243^6+43^6=338^6+177^6+168^6+95^6
(51)
351^6+265^6+221^6=336^6+309^6+169^6+73^6
(52)
365^6+137^6+126^6=360^6+234^6+175^6+133^6
(53)
360^6+265^6+200^6=336^6+318^6+212^6+169^6
(54)
348^6+325^6+36^6=357^6+276^6+276^6+82^6
(55)
373^6+288^6+104^6=363^6+292^6+266^6+120^6
(56)
386^6+113^6+62^6=378^6+260^6+209^6+88^6
(57)

(Lander et al. 1967, Ekl 1998).

Moessner (1947) gave three parametric solutions to the 6.4.4 equation. The smallest 6.4.4 solution is

2^6+2^6+9^6+9^6=3^6+5^6+6^6+10^6
(58)

(Rao 1934, Lander et al. 1967). The smallest 6.4.4.4 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
(59)

(Lander et al. 1967).

Moessner and Gloden (1944) give the 6.7.8 solution

32^6+31^6+23^6+22^6+13^6+6^6+5^6 =33^6+28^6+27^6+20^6+11^6+10^6+2^6+1^6.
(60)

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References

Ekl, R. L. "Equal Sums of Four Seventh Powers." Math. Comput. 65, 1755-1756, 1996.Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.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.Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." http://euler.free.fr.Meyrignac, J.-C. "Description of Resta's Algorithm." http://euler.free.fr/how.htm.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.Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. École Polytech. de Timisoara 11, 196-219, 1944.Rao, S. K. "On Sums of Sixth Powers." J. London Math. Soc. 9, 172-173, 1934.Resta, G. and Meyrignac, J.-C. "The Smallest Solutions to the Diophantine Equation x^6+y^6=a^6+b^6+c^6+d^6+e^6." Math. Comput. 72, 1051-1054, 2003.Update a linkResta, G. "New Results on Equal Sums of Sixth Powers." Instituto di Matematica Computazionale, Pisa, Italy. April 1999. http://www.chez.com/powersum/Tr-b4-08.zip

Cite this as:

Weisstein, Eric W. "Diophantine Equation--6th Powers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiophantineEquation6thPowers.html

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