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Diophantine Equation--10th Powers

The 10.1.2 equation

A^(10)=B^(10)+C^(10)
(1)

is a special case of Fermat's last theorem with n=10, and so has no solution. No 10.1.n solutions are known with n<13. A 10.1.13 solution is

228^(10)=210^(10)+204^(10)+187^(10)+179^(10)+128^(10)+122^(10)+85^(10)+73^(10)+59^(10)+57^(10)+49^(10)+13^(10)+6^(10)
(2)

(S. Chase). The smallest 10.1.15 solution is

100^(10)+94^(10)+91^(10)+2·77^(10)+76^(10)+63^(10)+62^(10)+52^(10)+45^(10)+35^(10)+33^(10)+16^(10)+10^(10)+1^(10)=108^(10)
(3)

(J.-C. Meyrignac 1999). The smallest 10.1.22 solution is

33^(10)=2·30^(10)+2·26^(10)+23^(10)+21^(10)+19^(10)+18^(10) +2·13^(10)+2·12^(10)+5·10^(10)+2·9^(10)+7^(10)+6^(10)+3^(10)
(4)

(Ekl 1998). The smallest 10.1.23 solution is

5·1^(10)+2^(10)+3^(10)+6^(10)+6·7^(10)+4·9^(10)+10^(10)+2·12^(10)+13^(10)+14^(10)=15^(10)
(5)

(Lander et al. 1967).

10.2.12 solutions include

135^(10)+55^(10)=129^(10)+115^(10)+105^(10)+103^(10)+83^(10)+80^(10)+71^(10)+71^(10)+51^(10)+47^(10)+15^(10)+12^(10)
(6)
112^(10)+99^(10)=109^(10)+103^(10)+83^(10)+79^(10)+72^(10)+59^(10)+59^(10)+52^(10)+20^(10)+15^(10)+5^(10)+5^(10)
(7)

(V. Pliousnine 2000, N. Kuosa 2000). The smallest 10.2.13 solution is

51^(10)+32^(10)=49^(10)+43^(10)+41^(10)+37^(10)+28^(10)+26^(10)+25^(10)+15^(10)+10^(10)+10^(10)+9^(10)+5^(10)+3^(10).
(8)

The smallest 10.2.15 solution is

35^(10)+3^(10)=33^(10)+32^(10)+24^(10)+21^(10)+2·20^(10)+3·13^(10)+12^(10)+11^(10)+9^(10)+7^(10)+2·1^(10)
(9)

(Ekl 1998). The smallest 10.2.19 solution is

5·2^(10)+5^(10)+6^(10)+10^(10)+6·11^(10)+2·12^(10)+3·15^(10)=9^(10)+17^(10)
(10)

(Lander et al. 1967). A 10.3.11 solution is

385^(10)+209^(10)+88^(10)=368^(10)+318^(10)+304^(10)+293^(10)+285^(10)+228^(10)+216^(10)+184^(10)+76^(10)+64^(10)+12^(10)
(11)

(J. Wroblewski 2002). A 10.3.12 solution is

120^(10)+44^(10)+22^(10)=116^(10)+102^(10)+90^(10)+85^(10)+65^(10)+51^(10)+51^(10)+41^(10)+37^(10)+23^(10)+5^(10)+2^(10)
(12)

(T. Nolan 2000). The smallest 10.3.13 solution is

46^(10)+32^(10)+22^(10)=43^(10)+43^(10)+27^(10)+26^(10)+17^(10)+16^(10)+12^(10)+9^(10)+9^(10)+6^(10)+4^(10)+3^(10)+3^(10).
(13)

The smallest 10.3.14 solution is

30^(10)+28^(10)+4^(10)=31^(10)+23^(10)+2·20^(10)+2·17^(10)+16^(10)+10^(10)+3·9^(10)+5^(10)+2·2^(10)
(14)

(Ekl 1998). The smallest 10.3.24 solution is

1^(10)+2^(10)+3^(10)+10·4^(10)+7^(10)+7·8^(10)+10^(10)+12^(10)+16^(10)=11^(10)+2·15^(10)
(15)

(Lander et al. 1967).

A 10.4.9 solution is

1723^(10)+1477^(10)+1040^(10)+246^(10)=1628^(10)+1542^(10)+1500^(10)+1221^(10)+1144^(10)+1130^(10)+1093^(10)+550^(10)+110^(10)
(16)

(J. Wroblewski 2002). 10.4.10 solutions include

797^(10)+260^(10)+103^(10)+75^(10)=748^(10)+704^(10)+646^(10)
(17)
+572^(10)+541^(10)+392^(10)+352^(10)+323^(10)+264^(10)+143^(10)
(18)
(19)
871^(10)+400^(10)+362^(10)+89^(10)=836^(10)+726^(10)+680^(10)
(20)
+638^(10)+638^(10)+462^(10)+389^(10)+218^(10)+99^(10)+34^(10)
(21)
(22)
969^(10)+521^(10)+317^(10)+114^(10)=902^(10)+882^(10)+759^(10)
(23)
+654^(10)+605^(10)+594^(10)+410^(10)+297^(10)+44^(10)+16^(10)
(24)
(25)
989^(10)+853^(10)+202^(10)+64^(10)=924^(10)+878^(10)+855^(10)
(26)
+784^(10)+770^(10)+548^(10)+506^(10)+352^(10)+231^(10)+22^(10)
(27)
(28)
992^(10)+657^(10)+181^(10)+75^(10)=946^(10)+842^(10)+829^(10)
(29)
+660^(10)+638^(10)+583^(10)+174^(10)+155^(10)+110^(10)+88^(10)
(30)
(31)
995^(10)+845^(10)+801^(10)+245^(10)=974^(10)+891^(10)+822^(10)
(32)
+660^(10)+539^(10)+539^(10)+502^(10)+308^(10)+286^(10)+177^(10)
(33)
(34)
919^(10)+855^(10)+613^(10)+586^(10)=924^(10)+825^(10)+702^(10)
(35)
+660^(10)+585^(10)+506^(10)+459^(10)+374^(10)+242^(10)+42^(10)
(36)
(37)
799^(10)+749^(10)+103^(10)+14^(10)=814^(10)+660^(10)+649^(10)
(38)
+583^(10)+448^(10)+386^(10)+330^(10)+197^(10)+44^(10)+24^(10)
(39)
(40)
953^(10)+799^(10)+213^(10)+188^(10)=885^(10)+836^(10)+825^(10)
(41)
+748^(10)+724^(10)+638^(10)+577^(10)+566^(10)+528^(10)+528^(10)
(42)
(43)
767^(10)+713^(10)+281^(10)+186^(10)=795^(10)+539^(10)+502^(10)
(44)
+440^(10)+425^(10)+330^(10)+282^(10)+264^(10)+44^(10)+22^(10)
(45)
(46)
962^(10)+529^(10)+310^(10)+9^(10)=911^(10)+880^(10)+616^(10)
(47)
+462^(10)+316^(10)+242^(10)+169^(10)+154^(10)+142^(10)+22^(10).
(48)

(J. Wroblewski 2002). A 10.4.11 solution is

264^(10)+209^(10)+99^(10)+66^(10)=252^(10)+240^(10)+196^(10)+184^(10)+150^(10)+140^(10)+81^(10)+76^(10)+62^(10)+56^(10)+29^(10)
(49)

(S. Chase). The 10.4.12 equation has solution

51^(10)+49^(10)+43^(10)+39^(10)+29^(10)+28^(10)+2·17^(10)+16^(10)+13^(10)+7^(10)+4^(10)=53^(10)+244^(10)+22^(10)
(50)

(E. Bainville 1999). The smallest 10.4.15 solution is

4·23^(10)=26^(10)+5·18^(10)+3·17^(10)+15^(10)+12^(10)+6^(10)+3·4^(10)
(51)

(Ekl 1998). The smallest 10.4.23 solution is

5·1^(10)+2·2^(10)+2·3^(10)+4^(10)+4·6^(10)+3·7^(10)+8^(10)+2·10^(10)+2·14^(10)+15^(10)=3·11^(10)+16^(10)
(52)

(Lander et al. 1967).

The smallest 10.5.16 solutions are

4·1^(10)+2^(10)+2·4^(10)+6^(10)+2·12^(10)+5·13^(10)+15^(10)=2·3^(10)+8^(10)+14^(10)+16^(10)
(53)
20^(10)+11^(10)+8^(10)+3^(10)+1^(10)=2·18^(10)+17^(10)+16^(10)+10^(10)+2·7^(10)+6·4^(10)+2·2^(10)
(54)

(Lander et al. 1967, Ekl 1998).

The smallest 10.6.6 solution is

95^(10)+71^(10)+32^(10)+28^(10)+25^(10)+16^(10) =92^(10)+85^(10)+34^(10)+34^(10)+23^(10)+5^(10).
(55)

The smallest 10.6.16 solution is

18^(10)+12^(10)+11^(10)+10^(10)+3^(10)+2^(10) =17^(10)+16^(10)+4·13^(10)+4·7^(10)+4·6^(10)+5^(10)+4^(10)
(56)

(Ekl 1998). The smallest 10.6.27 solution is

1^(10)+4·3^(10)+2·4^(10)+2·5^(10)+7·6^(10)+9·7^(10)+10^(10)+13^(10)=2·2^(10)+8^(10)+11^(10)+2·12^(10)
(57)

(Lander et al. 1967).

The smallest 10.7.7 solutions are

38^(10)+33^(10)+26^(10)+26^(10)+15^(10)+8^(10)+1^(10)
(58)
=36^(10)+35^(10)+32^(10)+29^(10)+24^(10)+23^(10)+22^(10)
(59)
68^(10)+61^(10)+55^(10)+32^(10)+31^(10)+28^(10)+1^(10)
(60)
=67^(10)+64^(10)+49^(10)+44^(10)+23^(10)+20^(10)+17^(10)
(61)

(Lander et al. 1967, Ekl 1998).

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References

Ekl, R. L. "New Results in Equal Sums of Like Powers." Math. Comput. 67, 1309-1315, 1998.Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446-459, 1967.Meyrignac, J.-C. "Computing Minimal Equal Sums of Like Powers." http://euler.free.fr.

Referenced on Wolfram|Alpha

Diophantine Equation--10th Powers

Cite this as:

Weisstein, Eric W. "Diophantine Equation--10th Powers." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DiophantineEquation10thPowers.html

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