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The Nine Digits Page 8
with some Ten Digits (pandigital) exceptions
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When I use the term ninedigital in these articles I always refer to a strictly zeroless pandigital (digits from 1 to 9 each appearing just once).

Eight Page

  Topic 8.2   [ August 24, 2025 ]
Nine- and pandigital's equal to the sum of two, three and four positive and/or negative fifthpowers.

Case of ninedigitals Case of pandigitals

One beautiful solution with only  TWO  fifthpowers !
It is the most privileged ninedigital.
And only the digits 0 and 5 are used, five in total.
The difference between the two terms \(50-5\) is \(45\) which is the digitsum of any nine- or pandigital.

\(\Large{\bbox[lightyellow,5px, border:1px green solid]{{\color{blue}{312496875}}={\color{purple}{50^5-5^5}}}}\)


Ninedigital solutions with  THREE  fifthpowers. Here is a complete list, one column with positive terms and another with
mixed signs terms. Do you know how many of both kinds exist ?
I counted  \(15\)  solutions for the case of positive terms only. Let me name them the second most privileged ninedigitals.
And there are  \(46\)  solutions for the case with mixed signs terms. These are the third most privileged ninedigitals.

\({\color{blueviolet}{128754693}}=21^5+34^5+38^5\)

\({\color{blueviolet}{192543867}}=10^5+34^5+43^5\)

\({\color{blueviolet}{247963518}}=30^5+33^5+45^5\)

\({\color{blueviolet}{296453871}}=32^5+32^5+47^5\)

\({\color{blueviolet}{312749856}}=4^5+12^5+50^5\)

\({\color{blueviolet}{418295736}}=3^5+10^5+53^5\)

\({\color{blueviolet}{451782693}}=8^5+32^5+53^5\)

\({\color{blueviolet}{453169782}}=17^5+32^5+53^5\)

\({\color{blueviolet}{564732189}}=2^5+45^5+52^5\)

\({\color{blueviolet}{643752198}}=9^5+45^5+54^5\)

\({\color{blueviolet}{783291456}}=14^5+22^5+60^5\)

\({\color{blueviolet}{823196475}}=11^5+34^5+60^5\)

\({\color{blueviolet}{832679145}}=36^5+41^5+58^5\)

\({\color{blueviolet}{854361927}}=1^5+25^5+61^5\)

\({\color{blueviolet}{862319745}}=1^5+46^5+58^5\)

 

\({\color{maroon}{146932587}}=(-7)^5+(-9)^5+43^5\)

\({\color{maroon}{157286493}}=(-19)^5+(-22)^5+44^5\)

\({\color{maroon}{157324869}}=(-22)^5+(-75)^5+76^5\)

\({\color{maroon}{184279536}}=3^5+(-12)^5+45^5\)

\({\color{maroon}{184523976}}=(-4)^5+(-5)^5+45^5\)

\({\color{maroon}{184527639}}=(-3)^5+(-3)^5+45^5\)

\({\color{maroon}{214937856}}=(-98)^5+(-100)^5+114^5\)

\({\color{maroon}{249673851}}=11^5+(-42)^5+52^5\)

\({\color{maroon}{253416879}}=8^5+(-17)^5+48^5\)

\({\color{maroon}{268794351}}=14^5+(-49)^5+56^5\)

\({\color{maroon}{274315896}}=(-93)^5+(-108)^5+117^5\qquad\)

\({\color{maroon}{278931456}}=(-28)^5+(-60)^5+64^5\)

\({\color{maroon}{293854617}}=25^5+(-72)^5+74^5\)

\({\color{maroon}{316894257}}=(-15)^5+22^5+50^5\)

\({\color{maroon}{318756249}}=(-1)^5+(-45)^5+55^5\)

\({\color{maroon}{352714968}}=43^5+(-51)^5+56^5\)

\({\color{maroon}{364921875}}=(-10)^5+35^5+50^5\)

\({\color{maroon}{381675249}}=(-20)^5+40^5+49^5\)

\({\color{maroon}{413729568}}=(-2)^5+(-34)^5+54^5\)

\({\color{maroon}{429856317}}=(-21)^5+(-37)^5+55^5\)

\({\color{maroon}{473826915}}=(-18)^5+42^5+51^5\)

\({\color{maroon}{531962784}}=36^5+(-38)^5+56^5\)

\({\color{maroon}{539726481}}=17^5+(-70)^5+74^5\)

\({\color{maroon}{542319768}}=(-10)^5+33^5+55^5\)

\({\color{maroon}{567914832}}=(-4)^5+(-43)^5+59^5\)

\({\color{maroon}{589324167}}=100^5+103^5+(-116)^5\)

\({\color{maroon}{623495718}}=39^5+(-54)^5+63^5\)

\({\color{maroon}{629835417}}=53^5+(-55)^5+59^5\)

\({\color{maroon}{647931285}}=8^5+(-62)^5+69^5\)

\({\color{maroon}{654871392}}=24^5+(-42)^5+60^5\)

\({\color{maroon}{712856493}}=(-16)^5+(-42)^5+61^5\)

\({\color{maroon}{725193468}}=41^5+(-61)^5+68^5\)

\({\color{maroon}{734925168}}=(-21)^5+(-59)^5+68^5\)

\({\color{maroon}{738245619}}=(-29)^5+40^5+58^5\)

\({\color{maroon}{768954132}}=(-23)^5+36^5+59^5\)

\({\color{maroon}{781435269}}=(-12)^5+21^5+60^5\)

\({\color{maroon}{786431592}}=3^5+(-60)^5+69^5\)

\({\color{maroon}{786439125}}=6^5+(-60)^5+69^5\)

\({\color{maroon}{854971326}}=(-44)^5+53^5+57^5\)

\({\color{maroon}{856247193}}=(-12)^5+48^5+57^5\)

\({\color{maroon}{923684157}}=(-40)^5+(-70)^5+77^5\)

\({\color{maroon}{928653417}}=24^5+(-87)^5+90^5\)

\({\color{maroon}{948172635}}=(-21)^5+(-72)^5+78^5\)

\({\color{maroon}{954327681}}=2^5+(-46)^5+65^5\)

\({\color{maroon}{983417625}}=(-10)^5+(-39)^5+64^5\)

\({\color{maroon}{983517624}}=(-1)^5+(-39)^5+64^5\)


Ninedigital solutions with  FOUR  fifthpowers.
At the left the first few and last of  \(191\)  solutions with positive terms and
at the right a batch from so many solutions with mixed signs terms with \(z\leqslant2000\).

\({\color{maroon}{124398756}}=12^5+13^5+32^5+39^5\)

\({\color{maroon}{127436958}}=10^5+27^5+35^5+36^5\)

\({\color{maroon}{128375469}}=1^5+29^5+31^5+38^5\)

\({\color{maroon}{128463597}}=24^5+25^5+29^5+39^5\)

\({\color{maroon}{128635749}}=14^5+19^5+25^5+41^5\)

\({\color{maroon}{128754936}}=3^5+21^5+34^5+38^5\)

\({\color{maroon}{129346875}}=5^5+30^5+35^5+35^5\)

\({\color{maroon}{134597268}}=11^5+25^5+34^5+38^5\)

\({\color{maroon}{134957286}}=3^5+20^5+35^5+38^5\)

\({\color{maroon}{143962785}}=14^5+18^5+33^5+40^5\)

\({\color{maroon}{146938275}}=18^5+19^5+26^5+42^5\)

\({\color{maroon}{149268537}}=23^5+25^5+28^5+41^5\)

⋮

\({\color{maroon}{935871264}}=17^5+29^5+37^5+61^5\)

\({\color{maroon}{942783156}}=2^5+24^5+39^5+61^5\)

\({\color{maroon}{945127368}}=6^5+29^5+43^5+60^5\)

\({\color{maroon}{948237165}}=10^5+17^5+51^5+57^5\)

\({\color{maroon}{952164387}}=4^5+19^5+32^5+62^5\)

\({\color{maroon}{952681743}}=21^5+25^5+49^5+58^5\)

\({\color{maroon}{963514782}}=7^5+16^5+54^5+55^5\)

\({\color{maroon}{968572431}}=5^5+30^5+47^5+59^5\)

\({\color{maroon}{973825146}}=7^5+22^5+35^5+62^5\)

\({\color{maroon}{978421356}}=4^5+25^5+35^5+62^5\)

\({\color{maroon}{982571634}}=20^5+21^5+42^5+61^5\)

\({\color{maroon}{984516273}}=12^5+34^5+49^5+58^5\)

\({\color{blue}{123456789}}=v^5+w^5+x^5+y^5+z^5\)

\({\color{maroon}{148596732}}=229^5+(-290)^5+(-378)^5+391^5\)

\({\color{maroon}{172359648}}=(-456)^5+(-550)^5+(-1366)^5+1370^5\)

\({\color{maroon}{176839542}}=(-222)^5+244^5+279^5+(-289)^5\)

\({\color{maroon}{182365974}}=(-133)^5+199^5+253^5+(-265)^5\)

\({\color{maroon}{183946257}}=710^5+977^5+1066^5+(-1196)^5\)

\({\color{maroon}{293754681}}=(-69)^5+(-171)^5+(-183)^5+204^5\)

\({\color{maroon}{317846925}}=46^5+93^5+134^5+(-138)^5\)

\({\color{maroon}{327149856}}=(-12)^5+82^5+112^5+(-116)^5\)

\({\color{maroon}{416985723}}=(-36)^5+139^5+174^5+(-184)^5\)

\({\color{maroon}{521438976}}=(-128)^5+166^5+308^5+(-310)^5\)

\({\color{maroon}{538196427}}=(-72)^5+186^5+187^5+(-214)^5\)

\({\color{maroon}{683215975}}=34^5+81^5+154^5+(-155)^5\)

\({\color{maroon}{741258396}}=282^5+(-345)^5+(-508)^5+517^5\)

\({\color{maroon}{798651423}}=(-141)^5+344^5+457^5+(-477)^5\)

\({\color{maroon}{816379425}}=(-63)^5+(-106)^5+(-164)^5+168^5\)

\({\color{maroon}{873146925}}=(-39)^5+(-100)^5+(-151)^5+155^5\)

\({\color{maroon}{912386574}}=520^5+(-600)^5+(-861)^5+875^5\)

\({\color{maroon}{936587214}}=(-123)^5+(-140)^5+(-168)^5+185^5\)

\({\color{maroon}{938574216}}=(-201)^5+209^5+288^5+(-290)^5\)

\({\color{maroon}{985134726}}=(-40)^5+146^5+185^5+(-195)^5\)

\({\color{maroon}{985631427}}=(-182)^5+(-198)^5+(-269)^5+286^5\)

\({\color{blue}{987654321}}=v^5+w^5+x^5+y^5+z^5\)

 

One beautiful solution with only  TWO  fifthpowers !
It is the most privileged pandigital.
And only digits 0,5,6,7 and 8 are used, five in total.
The difference between the two terms \(78-60\) is \(18\) which divides our pandigital.

\(\Large{\bbox[mintcream,5px, border:1px green solid]{{\color{blue}{2109574368}}={\color{purple}{78^5-60^5}}}}\)


Pandigital solutions with  THREE  fifthpowers. Here is a complete list, one column with positive terms and another with
mixed signs terms. Do you know how many of both kinds exist ?
I counted  \(55\)  solutions for the case of positive terms only. Let me name them the second most privileged pandigitals.
And there are  \(199\)  solutions for the case with mixed signs terms. These are the third most privileged pandigitals.

\({\color{maroon}{1286375904}}=33^5+48^5+63^5\qquad\)

\({\color{maroon}{1396087452}}=15^5+56^5+61^5\)

\({\color{maroon}{1438750269}}=18^5+45^5+66^5\)

\({\color{maroon}{1642938507}}=26^5+59^5+62^5\)

\({\color{maroon}{1752849306}}=39^5+50^5+67^5\)

\({\color{maroon}{1846509723}}=5^5+49^5+69^5\)

\({\color{maroon}{1974053268}}=3^5+33^5+72^5\)

\({\color{maroon}{2017596384}}=56^5+56^5+62^5\)

\({\color{maroon}{2195740638}}=42^5+59^5+67^5\)

\({\color{maroon}{2318094657}}=52^5+60^5+65^5\)

\({\color{maroon}{2537418069}}=5^5+18^5+76^5\)

\({\color{maroon}{2561430897}}=49^5+59^5+69^5\)

\({\color{maroon}{2637805941}}=8^5+64^5+69^5\)

\({\color{maroon}{3018624975}}=15^5+42^5+78^5\)

\({\color{maroon}{3082145967}}=18^5+20^5+79^5\)

\({\color{maroon}{3198204657}}=49^5+52^5+76^5\)

\({\color{maroon}{3284067951}}=16^5+46^5+79^5\)

\({\color{maroon}{3421869507}}=11^5+59^5+77^5\)

\({\color{maroon}{3692780145}}=8^5+46^5+81^5\)

\({\color{maroon}{3719528640}}=12^5+26^5+82^5\)

\({\color{maroon}{3754820169}}=16^5+70^5+73^5\)

\({\color{maroon}{3926481075}}=55^5+67^5+73^5\)

\({\color{maroon}{4035671982}}=50^5+67^5+75^5\)

\({\color{maroon}{4059217638}}=37^5+49^5+82^5\)

\({\color{maroon}{4196527380}}=9^5+27^5+84^5\)

\({\color{maroon}{4312869705}}=9^5+42^5+84^5\)

\({\color{maroon}{4320971865}}=26^5+57^5+82^5\)

\({\color{maroon}{4539687120}}=36^5+63^5+81^5\)

\({\color{maroon}{4628017593}}=23^5+45^5+85^5\)

\({\color{maroon}{4930621875}}=45^5+75^5+75^5\)

\({\color{maroon}{4985027631}}=9^5+15^5+87^5\)

\({\color{maroon}{4986753201}}=6^5+49^5+86^5\)

\({\color{maroon}{5320687419}}=36^5+75^5+78^5\)

\({\color{maroon}{5406781392}}=48^5+59^5+85^5\)

\({\color{maroon}{5429136087}}=23^5+76^5+78^5\)

\({\color{maroon}{5893617024}}=54^5+66^5+84^5\)

\({\color{maroon}{6250347819}}=71^5+73^5+75^5\)

\({\color{maroon}{6405237918}}=3^5+44^5+91^5\)

\({\color{maroon}{6894057213}}=22^5+77^5+84^5\)

\({\color{maroon}{7049258613}}=59^5+67^5+87^5\)

\({\color{maroon}{7069352841}}=9^5+78^5+84^5\)

\({\color{maroon}{7215640983}}=56^5+70^5+87^5\)

\({\color{maroon}{7239854601}}=9^5+76^5+86^5\)

\({\color{maroon}{7359204861}}=13^5+68^5+90^5\)

\({\color{maroon}{7386491025}}=70^5+74^5+81^5\)

\({\color{maroon}{7398641250}}=48^5+77^5+85^5\)

\({\color{maroon}{7498215360}}=18^5+74^5+88^5\)

\({\color{maroon}{7594382016}}=14^5+48^5+94^5\)

\({\color{maroon}{7923840651}}=5^5+81^5+85^5\)

\({\color{maroon}{8237045961}}=73^5+78^5+80^5\)

\({\color{maroon}{8615970432}}=4^5+31^5+97^5\)

\({\color{maroon}{9260384157}}=77^5+80^5+80^5\)

\({\color{maroon}{9368140257}}=20^5+60^5+97^5\)

\({\color{maroon}{9603821457}}=72^5+81^5+84^5\)

\({\color{maroon}{9825170643}}=36^5+48^5+99^5\)

 

\({\color{maroon}{1054286793}}=62^5+(-72)^5+73^5\)

\({\color{maroon}{1054926873}}=(-66)^5+(-80)^5+89^5\)

\({\color{maroon}{1059687234}}=(-24)^5+(-49)^5+67^5\)

\({\color{maroon}{1075349826}}=(-19)^5+21^5+64^5\)

\({\color{maroon}{1089437526}}=(-30)^5+(-75)^5+81^5\)

\({\color{maroon}{1246358907}}=(-18)^5+(-21)^5+66^5\)

\({\color{maroon}{1250693487}}=38^5+(-49)^5+68^5\)

\({\color{maroon}{1293045768}}=(-57)^5+(-63)^5+78^5\)

\({\color{maroon}{1362874950}}=(-25)^5+52^5+63^5\)

\({\color{maroon}{1394568702}}=(-15)^5+56^5+61^5\)

\({\color{maroon}{1397048256}}=100^5+142^5+(-146)^5\)

\({\color{maroon}{1432869507}}=(-13)^5+(-68)^5+78^5\)

\({\color{maroon}{1468279350}}=(-5)^5+27^5+68^5\)

\({\color{maroon}{1524836907}}=(-9)^5+(-33)^5+69^5\)

\({\color{maroon}{1532906847}}=(-1)^5+54^5+64^5\)

\({\color{maroon}{1548923067}}=(-15)^5+(-27)^5+69^5\)

\({\color{maroon}{1568720493}}=(-25)^5+53^5+65^5\)

\({\color{maroon}{1569027843}}=(-15)^5+(-55)^5+73^5\)

\({\color{maroon}{1620394875}}=11^5+(-36)^5+70^5\)

\({\color{maroon}{1639254078}}=(-9)^5+(-44)^5+71^5\)

\({\color{maroon}{1642708539}}=52^5+(-53)^5+70^5\)

\({\color{maroon}{1683790254}}=(-57)^5+(-57)^5+78^5\)

\({\color{maroon}{1684907325}}=(-51)^5+60^5+66^5\)

\({\color{maroon}{1690432857}}=(-8)^5+25^5+70^5\)

\({\color{maroon}{1692870543}}=56^5+(-72)^5+79^5\)

\({\color{maroon}{1750389264}}=(-3)^5+(-45)^5+72^5\)

\({\color{maroon}{1768392405}}=(-48)^5+54^5+69^5\)

\({\color{maroon}{1846305792}}=8^5+(-96)^5+100^5\)

\({\color{maroon}{1857964032}}=(-56)^5+(-84)^5+92^5\)

\({\color{maroon}{1890653724}}=(-21)^5+(-63)^5+78^5\)

\({\color{maroon}{1924835076}}=57^5+(-69)^5+78^5\)

\({\color{maroon}{1932408765}}=(-8)^5+(-19)^5+72^5\)

\({\color{maroon}{1936047825}}=(-15)^5+18^5+72^5\)

\({\color{maroon}{1936784520}}=31^5+(-44)^5+73^5\)

\({\color{maroon}{1957328064}}=(-18)^5+30^5+72^5\)

\({\color{maroon}{2048671593}}=(-10)^5+(-30)^5+73^5\)

\({\color{maroon}{2071354869}}=15^5+(-19)^5+73^5\)

\({\color{maroon}{2341608795}}=11^5+(-102)^5+106^5\)

\({\color{maroon}{2387146950}}=(-12)^5+27^5+75^5\)

\({\color{maroon}{2405371968}}=14^5+(-42)^5+76^5\)

\({\color{maroon}{2437950681}}=42^5+(-80)^5+89^5\)

\({\color{maroon}{2450891367}}=39^5+(-62)^5+80^5\)

\({\color{maroon}{2506987143}}=7^5+(-52)^5+78^5\)

\({\color{maroon}{2518439607}}=(-33)^5+45^5+75^5\)

\({\color{maroon}{2583761904}}=(-22)^5+(-67)^5+83^5\)

\({\color{maroon}{2704895613}}=4^5+(-18)^5+77^5\)

\({\color{maroon}{2740165893}}=(-2)^5+(-43)^5+78^5\)

\({\color{maroon}{2785601349}}=34^5+(-43)^5+78^5\)

\({\color{maroon}{2840931576}}=(-9)^5+65^5+70^5\)

\({\color{maroon}{2891645703}}=(-44)^5+66^5+71^5\)

\({\color{maroon}{2917506843}}=(-11)^5+61^5+73^5\)

\({\color{maroon}{2946180357}}=90^5+117^5+(-120)^5\)

\({\color{maroon}{2970654831}}=(-21)^5+57^5+75^5\)

\({\color{maroon}{2983507461}}=38^5+(-75)^5+88^5\)

\({\color{maroon}{3018254976}}=(-32)^5+51^5+77^5\)

\({\color{maroon}{3052941867}}=(-25)^5+(-27)^5+79^5\)

\({\color{maroon}{3107628945}}=16^5+(-52)^5+81^5\)

\({\color{maroon}{3124879650}}=(-10)^5+53^5+77^5\)

\({\color{maroon}{3149206875}}=11^5+(-110)^5+114^5\)

\({\color{maroon}{3169458720}}=(-9)^5+(-87)^5+96^5\)

\({\color{maroon}{3207459168}}=5^5+(-37)^5+80^5\)

\({\color{maroon}{3276840951}}=(-9)^5+10^5+80^5\)

\({\color{maroon}{3279401568}}=(-17)^5+(-46)^5+81^5\)

\({\color{maroon}{3298014567}}=(-57)^5+70^5+74^5\)

\({\color{maroon}{3467198250}}=5^5+(-59)^5+84^5\)

\({\color{maroon}{3486725109}}=(-3)^5+(-9)^5+81^5\)

\({\color{maroon}{3487960512}}=163^5+386^5+(-387)^5\)

\({\color{maroon}{3604581792}}=24^5+(-70)^5+88^5\)

\({\color{maroon}{3604971582}}=9^5+(-123)^5+126^5\)

\({\color{maroon}{3612409758}}=110^5+187^5+(-189)^5\)

\({\color{maroon}{3647812950}}=189^5+265^5+(-274)^5\)

\({\color{maroon}{3648579021}}=142^5+322^5+(-323)^5\)

\({\color{maroon}{3690472158}}=(-39)^5+(-58)^5+85^5\)

\({\color{maroon}{3704162958}}=(-15)^5+(-19)^5+82^5\)

\({\color{maroon}{3762049851}}=11^5+(-88)^5+98^5\)

\({\color{maroon}{3952740186}}=103^5+118^5+(-125)^5\)

\({\color{maroon}{3968257401}}=(-71)^5+(-154)^5+156^5\qquad\)

\({\color{maroon}{4017235968}}=8^5+(-44)^5+84^5\)

\({\color{maroon}{4018957632}}=10^5+(-53)^5+85^5\)

\({\color{maroon}{4073869251}}=46^5+(-109)^5+114^5\)

\({\color{maroon}{4095132768}}=8^5+(-90)^5+100^5\)

\({\color{maroon}{4208761593}}=(-54)^5+(-62)^5+89^5\)

\({\color{maroon}{4253781069}}=61^5+(-92)^5+100^5\)

\({\color{maroon}{4382516907}}=(-3)^5+(-57)^5+87^5\)

\({\color{maroon}{4520318976}}=(-56)^5+76^5+76^5\)

\({\color{maroon}{4568972013}}=(-61)^5+77^5+77^5\)

\({\color{maroon}{4572863910}}=(-57)^5+63^5+84^5\)

\({\color{maroon}{4576298301}}=45^5+(-50)^5+86^5\)

\({\color{maroon}{4582706931}}=49^5+(-143)^5+145^5\)

\({\color{maroon}{4583921076}}=(-88)^5+(-147)^5+151^5\)

\({\color{maroon}{4609182357}}=(-27)^5+62^5+82^5\)

\({\color{maroon}{4756921830}}=(-96)^5+(-141)^5+147^5\)

\({\color{maroon}{4789203651}}=(-72)^5+(-144)^5+147^5\)

\({\color{maroon}{4806952173}}=(-51)^5+59^5+85^5\)

\({\color{maroon}{4829530716}}=(-51)^5+63^5+84^5\)

\({\color{maroon}{4831972056}}=(-31)^5+(-105)^5+112^5\)

\({\color{maroon}{4851976032}}=(-28)^5+44^5+86^5\)

\({\color{maroon}{4852916307}}=90^5+90^5+(-93)^5\)

\({\color{maroon}{4892607315}}=(-44)^5+67^5+82^5\)

\({\color{maroon}{4928576301}}=(-48)^5+(-99)^5+108^5\)

\({\color{maroon}{4932075618}}=28^5+(-37)^5+87^5\)

\({\color{maroon}{4952637801}}=(-23)^5+48^5+86^5\)

\({\color{maroon}{5032914786}}=18^5+(-77)^5+95^5\)

\({\color{maroon}{5042981637}}=(-72)^5+(-107)^5+116^5\)

\({\color{maroon}{5086237914}}=(-13)^5+40^5+87^5\)

\({\color{maroon}{5176328409}}=(-39)^5+(-78)^5+96^5\)

\({\color{maroon}{5179648032}}=(-48)^5+66^5+84^5\)

\({\color{maroon}{5186023974}}=(-31)^5+60^5+85^5\)

\({\color{maroon}{5196374208}}=36^5+(-90)^5+102^5\)

\({\color{maroon}{5290634187}}=(-32)^5+(-62)^5+91^5\)

\({\color{maroon}{5290738641}}=78^5+(-111)^5+114^5\)

\({\color{maroon}{5346708192}}=110^5+124^5+(-132)^5\)

\({\color{maroon}{5371890642}}=(-27)^5+(-100)^5+109^5\)

\({\color{maroon}{5384206719}}=(-50)^5+63^5+86^5\)

\({\color{maroon}{5413867920}}=(-69)^5+(-107)^5+116^5\)

\({\color{maroon}{5421067389}}=(-70)^5+(-83)^5+102^5\)

\({\color{maroon}{5481320769}}=(-129)^5+(-137)^5+155^5\)

\({\color{maroon}{5642903718}}=75^5+(-120)^5+123^5\)

\({\color{maroon}{5709643182}}=(-48)^5+(-63)^5+93^5\)

\({\color{maroon}{5746218930}}=(-1)^5+(-61)^5+92^5\)

\({\color{maroon}{5761049832}}=(-15)^5+60^5+87^5\)

\({\color{maroon}{5780431269}}=(-12)^5+(-75)^5+96^5\)

\({\color{maroon}{5809614732}}=(-25)^5+(-110)^5+117^5\)

\({\color{maroon}{5871260493}}=(-26)^5+(-64)^5+93^5\)

\({\color{maroon}{5904867231}}=(-1)^5+(-8)^5+90^5\)

\({\color{maroon}{5934016782}}=104^5+157^5+(-159)^5\)

\({\color{maroon}{5943276018}}=(-15)^5+33^5+90^5\)

\({\color{maroon}{5974102386}}=(-67)^5+78^5+85^5\)

\({\color{maroon}{6095734218}}=33^5+(-87)^5+102^5\)

\({\color{maroon}{6179283450}}=(-3)^5+(-60)^5+93^5\)

\({\color{maroon}{6237845109}}=(-3)^5+(-19)^5+91^5\)

\({\color{maroon}{6239851740}}=17^5+(-18)^5+91^5\)

\({\color{maroon}{6241958370}}=(-4)^5+(-59)^5+93^5\)

\({\color{maroon}{6243719085}}=48^5+(-67)^5+94^5\)

\({\color{maroon}{6309127584}}=(-14)^5+37^5+91^5\)

\({\color{maroon}{6327981045}}=(-2)^5+(-84)^5+101^5\)

\({\color{maroon}{6350917482}}=17^5+(-71)^5+96^5\)

\({\color{maroon}{6391257408}}=(-14)^5+(-158)^5+160^5\)

\({\color{maroon}{6403128975}}=(-177)^5+(-228)^5+240^5\)

\({\color{maroon}{6457932108}}=115^5+137^5+(-144)^5\)

\({\color{maroon}{6512890374}}=(-51)^5+(-90)^5+105^5\)

\({\color{maroon}{6540817392}}=(-27)^5+75^5+84^5\)

\({\color{maroon}{6857923401}}=6^5+(-90)^5+105^5\)

\({\color{maroon}{6879531042}}=(-60)^5+73^5+89^5\)

\({\color{maroon}{6910724358}}=21^5+(-70)^5+97^5\)

\({\color{maroon}{6915042738}}=(-21)^5+72^5+87^5\)

\({\color{maroon}{7029146583}}=162^5+215^5+(-224)^5\)

\({\color{maroon}{7039145268}}=(-24)^5+39^5+93^5\)

\({\color{maroon}{7039185462}}=(-32)^5+41^5+93^5\)

\({\color{maroon}{7081452639}}=2^5+(-58)^5+95^5\)

\({\color{maroon}{7143829506}}=(-6)^5+77^5+85^5\)

\({\color{maroon}{7182365094}}=(-72)^5+(-129)^5+135^5\)

\({\color{maroon}{7230894516}}=(-21)^5+(-71)^5+98^5\)

\({\color{maroon}{7269485301}}=(-59)^5+(-84)^5+104^5\)

\({\color{maroon}{7319562480}}=74^5+(-81)^5+97^5\)

\({\color{maroon}{7340681295}}=3^5+(-99)^5+111^5\)

\({\color{maroon}{7364981052}}=30^5+(-99)^5+111^5\)

\({\color{maroon}{7425309618}}=3^5+(-50)^5+95^5\)

\({\color{maroon}{7526439810}}=(-42)^5+73^5+89^5\)

\({\color{maroon}{7536048192}}=(-66)^5+(-90)^5+108^5\)

\({\color{maroon}{7682419350}}=36^5+(-41)^5+95^5\)

\({\color{maroon}{7834592160}}=84^5+(-102)^5+108^5\)

\({\color{maroon}{7842951360}}=(-132)^5+(-162)^5+174^5\)

\({\color{maroon}{7863420951}}=(-24)^5+78^5+87^5\)

\({\color{maroon}{7945620318}}=(-12)^5+(-69)^5+99^5\)

\({\color{maroon}{7984236150}}=55^5+(-129)^5+134^5\)

\({\color{maroon}{8012475693}}=81^5+(-87)^5+99^5\)

\({\color{maroon}{8147653920}}=18^5+(-24)^5+96^5\)

\({\color{maroon}{8153694207}}=(-1)^5+(-8)^5+96^5\)

\({\color{maroon}{8153726490}}=(-3)^5+(-3)^5+96^5\)

\({\color{maroon}{8253941607}}=128^5+142^5+(-153)^5\)

\({\color{maroon}{8401297563}}=(-156)^5+(-213)^5+222^5\)

\({\color{maroon}{8436125907}}=(-8)^5+(-64)^5+99^5\)

\({\color{maroon}{8467093152}}=(-54)^5+(-64)^5+100^5\)

\({\color{maroon}{8602593174}}=51^5+(-66)^5+99^5\)

\({\color{maroon}{8602951743}}=(-1)^5+(-142)^5+146^5\)

\({\color{maroon}{8635107294}}=(-80)^5+(-107)^5+121^5\)

\({\color{maroon}{8723469150}}=60^5+(-69)^5+99^5\)

\({\color{maroon}{8932754601}}=(-45)^5+(-129)^5+135^5\)

\({\color{maroon}{9016482375}}=39^5+(-64)^5+100^5\)

\({\color{maroon}{9027385641}}=9^5+(-26)^5+98^5\)

\({\color{maroon}{9138064257}}=(-6)^5+56^5+97^5\)

\({\color{maroon}{9160745283}}=(-58)^5+(-210)^5+211^5\)

\({\color{maroon}{9204568317}}=18^5+(-115)^5+124^5\)

\({\color{maroon}{9275640318}}=(-37)^5+(-44)^5+99^5\)

\({\color{maroon}{9280716543}}=11^5+(-47)^5+99^5\)

\({\color{maroon}{9318027456}}=84^5+(-90)^5+102^5\)

\({\color{maroon}{9342508176}}=(-19)^5+(-44)^5+99^5\)

\({\color{maroon}{9365082174}}=83^5+(-181)^5+182^5\)

\({\color{maroon}{9410675832}}=(-1)^5+(-103)^5+116^5\)

\({\color{maroon}{9457032168}}=(-13)^5+53^5+98^5\)

\({\color{maroon}{9503467281}}=5^5+(-23)^5+99^5\)

\({\color{maroon}{9541073268}}=(-24)^5+33^5+99^5\)

\({\color{maroon}{9567014382}}=71^5+(-198)^5+199^5\)

\({\color{maroon}{9580176342}}=(-45)^5+48^5+99^5\)

\({\color{maroon}{9618240357}}=121^5+167^5+(-171)^5\)

\({\color{maroon}{9681243750}}=45^5+(-55)^5+100^5\)

\({\color{maroon}{9782134560}}=(-32)^5+(-176)^5+178^5\)

\({\color{maroon}{9783450612}}=(-54)^5+(-67)^5+103^5\)


Pandigital solutions with  FOUR  fifthpowers.
At the left the first few and last of  \(1022\)  solutions with positive terms (\(1\) pandigital has two solutions)
and at the right a batch of so many solutions with mixed signs terms with \(z\leqslant2000\).

\({\color{maroon}{1025783649}}=21^5+30^5+49^5+59^5\)

\({\color{maroon}{1027496835}}=16^5+42^5+51^5+56^5\)

\({\color{maroon}{1035467892}}=17^5+19^5+33^5+63^5\)

\({\color{maroon}{1037256849}}=44^5+45^5+45^5+55^5\)

\({\color{maroon}{1046935782}}=23^5+32^5+47^5+60^5\)

\({\color{maroon}{1047968523}}=19^5+33^5+39^5+62^5\)

\({\color{maroon}{1052938467}}=23^5+29^5+32^5+63^5\)

\({\color{maroon}{1067325498}}=19^5+40^5+54^5+55^5\)

\({\color{maroon}{1074923586}}=2^5+13^5+53^5+58^5\)

\({\color{maroon}{1075498362}}=14^5+42^5+47^5+59^5\)

\({\color{maroon}{1075684239}}=42^5+44^5+47^5+56^5\)

\({\color{maroon}{1075849632}}=26^5+34^5+40^5+62^5\)

⋮

\(\bbox[mintcream,3px,border:1px solid]{{\color{maroon}{1392508467}}\mathbf{\color{blue}{\;=\;}}3^5+28^5+54^5+62^5\mathbf{\color{blue}{\;=\;}}24^5+28^5+28^5+67^5}\)

Note that we have a multigrade here as
\(3+28+54+62\mathbf{\color{blue}{\;=\;}}147\mathbf{\color{blue}{\;=\;}}24+28+28+67\)

⋮

\({\color{maroon}{9765012348}}=9^5+12^5+48^5+99^5\)

\({\color{maroon}{9768314025}}=17^5+35^5+85^5+88^5\)

\({\color{maroon}{9784305126}}=48^5+62^5+75^5+91^5\)

\({\color{maroon}{9804612357}}=29^5+75^5+77^5+86^5\)

\({\color{maroon}{9805314267}}=1^5+33^5+84^5+89^5\)

\({\color{maroon}{9821506473}}=10^5+65^5+79^5+89^5\)

\({\color{maroon}{9842035761}}=43^5+55^5+68^5+95^5\)

\({\color{maroon}{9845627013}}=7^5+45^5+64^5+97^5\)

\({\color{maroon}{9846701523}}=4^5+30^5+50^5+99^5\)

\({\color{maroon}{9851037426}}=6^5+31^5+50^5+99^5\)

\({\color{maroon}{9852736014}}=37^5+57^5+80^5+90^5\)

\({\color{maroon}{9873562401}}=2^5+33^5+70^5+96^5\)

\({\color{blue}{123456789}}=v^5+w^5+x^5+y^5+z^5\)

\({\color{maroon}{1396752408}}=84^5+151^5+216^5+(-223)^5\)

\({\color{maroon}{1932640875}}=(-59)^5+(-167)^5+(-403)^5+404^5\)

\({\color{maroon}{1958047632}}=(-56)^5+(-83)^5+(-189)^5+190^5\)

\({\color{maroon}{2109574368}}=(-60)^5+78^5+889^5+(-889)^5\)

\({\color{maroon}{2983104576}}=400^5+(-494)^5+(-782)^5+792^5\)

\({\color{maroon}{3678419205}}=49^5+(-213)^5+(-220)^5+249^5\)

\({\color{maroon}{4198237056}}=(-276)^5+291^5+317^5+(-326)^5\)

\({\color{maroon}{4628517093}}=707^5+1128^5+1563^5+(-1625)^5\)

\({\color{maroon}{5793402186}}=(-241)^5+(-722)^5+(-1442)^5+1451^5\)

\({\color{maroon}{6204891375}}=267^5+(-528)^5+(-667)^5+703^5\)

\({\color{maroon}{6385701924}}=(-276)^5+(-433)^5+(-507)^5+550^5\)

\({\color{maroon}{8065713294}}=(-107)^5+238^5+297^5+(-314)^5\)

\({\color{maroon}{9186250743}}=153^5+(-297)^5+(-345)^5+372^5\)

\({\color{blue}{987654321}}=v^5+w^5+x^5+y^5+z^5\)


  Topic 8.1   [ August 24, 2025 ]
Nine- and pandigital's equal to the sum of five positive and/or negative fifthpowers. Building a longer and longer list of solutions.

Regarding the ninedigitals we know from Topic 4.5 that there is only one solution with five positive terms
so that when all these five terms are concatenated we get another nice ninedigital !

\(\Large{\bbox[lightyellow,5px,border:1px green solid]{{\color{blueviolet}{816725493}}={\color{green}{9}}^5+{\color{green}{17}}^5+{\color{green}{26}}^5+{\color{green}{43}}^5+{\color{green}{58}}^5~~\to~~{\color{green}{917264358}}~~\text{ is also ninedigital ! }}}\)

Case of ninedigitals Case of pandigitals

Firstly how many ninedigital solutions are there in general with five only positive terms ?
Here are already the first and last ten solutions from a total of  \(2246\)  :


\({\color{maroon}{123486957}}=1^5+4^5+19^5+22^5+41^5\)

\({\color{maroon}{123495687}}=7^5+22^5+24^5+24^5+40^5\)

\({\color{maroon}{123764895}}=12^5+28^5+28^5+31^5+36^5\)

\({\color{maroon}{123765489}}=15^5+21^5+27^5+27^5+39^5\)

\({\color{maroon}{123789564}}=2^5+5^5+6^5+32^5+39^5\)

\({\color{maroon}{123798564}}=1^5+5^5+7^5+32^5+39^5\)

\({\color{maroon}{123864597}}=9^5+19^5+19^5+31^5+39^5\)

\({\color{maroon}{124358697}}=4^5+4^5+14^5+24^5+41^5\)

\({\color{maroon}{124598376}}=9^5+18^5+31^5+32^5+36^5\)

\({\color{maroon}{124689357}}=2^5+19^5+27^5+31^5+38^5\)

⋮

Some ninedigitals can be expressed in two ways, some of which are multigrades.

The multigrades here happen when using exponent \(1\) & \(5\) and they sum up to equal values.

\(\bbox[mintcream,3px,border:1px solid]{{\color{maroon}{617492358}}\mathbf{\color{blue}{\;=\;}}16^5+25^5+29^5+46^5+52^5\mathbf{\color{blue}{\;=\;}}21^5+21^5+27^5+49^5+50^5}\)

\(16+25+29+46+52\mathbf{\color{blue}{\;=\;}}168\mathbf{\color{blue}{\;=\;}}21+21+27+49+50\)

\(\bbox[mintcream,3px,border:1px solid]{{\color{maroon}{764251389}}\mathbf{\color{blue}{\;=\;}}16^5+32^5+44^5+45^5+52^5\mathbf{\color{blue}{\;=\;}}23^5+31^5+33^5+51^5+51^5}\)

\(16+32+44+45+52\mathbf{\color{blue}{\;=\;}}189\mathbf{\color{blue}{\;=\;}}23+31+33+51+51\)

\(\bbox[mintcream,3px,border:1px solid]{{\color{maroon}{895762143}}\mathbf{\color{blue}{\;=\;}}2^5+38^5+41^5+49^5+53^5\mathbf{\color{blue}{\;=\;}}13^5+29^5+37^5+48^5+56^5}\)

\(2+38+41+49+53\mathbf{\color{blue}{\;=\;}}183\mathbf{\color{blue}{\;=\;}}13+29+37+48+56\)

⋮

\({\color{maroon}{985327164}}=13^5+17^5+37^5+50^5+57^5\)

\({\color{maroon}{986275413}}=4^5+30^5+47^5+47^5+55^5\)

\({\color{maroon}{986354172}}=15^5+22^5+22^5+42^5+61^5\)

\({\color{maroon}{986527413}}=4^5+4^5+16^5+37^5+62^5\)

\({\color{maroon}{986542173}}=1^5+7^5+16^5+37^5+62^5\)

\({\color{maroon}{986713542}}=5^5+38^5+49^5+50^5+50^5\)

\({\color{maroon}{987362451}}=20^5+35^5+41^5+50^5+55^5\)

\({\color{maroon}{987562134}}=11^5+17^5+39^5+51^5+56^5\)

\({\color{maroon}{987615432}}=3^5+12^5+18^5+37^5+62^5\)

\({\color{maroon}{987623415}}=4^5+24^5+28^5+54^5+55^5\)


Secondly I would like you to start a list of solutions with ninedigitals equal to five mixed signs terms.
Of course, thousands of ninedigitals will pop up with a sum of five fifthpowers. Therefore I will make a connecting
list going up from the smallest found ninedigital and going down from the largest found ninedigital.
Special solutions with e.g. a rare beauty or an interesting twist may also be reported.

\({\color{blue}{123456789}}=v^5+w^5+x^5+y^5+z^5\)

\({\color{maroon}{123458976}}=(-84)^5+(-228)^5+276^5+314^5+(-332)^5\)

\({\color{maroon}{123459687}}=11^5+87^5+(-102)^5+(-187)^5+188^5\)

\({\color{maroon}{123459876}}=(-214)^5+(-448)^5+532^5+553^5+(-597)^5\)

\({\color{maroon}{123469587}}=824^5+1091^5+(-1114)^5+1138^5+(-1162)^5\)

\({\color{maroon}{123497856}}=(-121)^5+(-493)^5+503^5+670^5+(-673)^5\)

\({\color{maroon}{123568974}}=152^5+(-445)^5+(-461)^5+(-497)^5+585^5\)

\({\color{maroon}{123589467}}=24^5+30^5+124^5+276^5+(-277)^5\)

\({\color{maroon}{123648975}}=(-19)^5+30^5+67^5+94^5+(-97)^5\)

\({\color{maroon}{123769458}}=71^5+(-476)^5+490^5+571^5+(-578)^5\)

\({\color{maroon}{123954876}}=(-29)^5+(-68)^5+(-87)^5+(-102)^5+112^5\)

⋮

\({\color{maroon}{986251374}}=253^5+494^5+1025^5+1095^5+(-1223)^5\)

\({\color{maroon}{987154632}}=55^5+217^5+(-239)^5+(-281)^5+290^5\)

\({\color{maroon}{987241563}}=12^5+(-71)^5+(-93)^5+(-138)^5+143^5\)

\({\color{maroon}{987516432}}=(-73)^5+101^5+(-199)^5+(-376)^5+379^5\)

\({\color{maroon}{987531624}}=(-195)^5+203^5+(-330)^5+(-336)^5+382^5\)

\({\color{maroon}{987621534}}=(-82)^5+(-176)^5+193^5+370^5+(-371)^5\)

\({\color{maroon}{987624153}}=98^5+729^5+1308^5+1440^5+(-1592)^5\)

\({\color{maroon}{987653142}}=143^5+295^5+(-352)^5+(-436)^5+452^5\)

\({\color{maroon}{987654231}}=(-228)^5+(-266)^5+311^5+420^5+(-426)^5\)

\({\color{blue}{987654321}}=v^5+w^5+x^5+y^5+z^5\)

 

Firstly how many pandigital solutions are there in general with five only positive terms ?
Here are already the first and last ten solutions from a total of  \(19183\)  :


\({\color{maroon}{1023459867}}=4^5+13^5+20^5+53^5+57^5\)

\({\color{maroon}{1023469587}}=13^5+28^5+34^5+41^5+61^5\)

\({\color{maroon}{1023469587}}=15^5+18^5+21^5+30^5+63^5\)

\({\color{maroon}{1023479568}}=14^5+16^5+29^5+51^5+58^5\)

\({\color{maroon}{1023685497}}=9^5+9^5+28^5+39^5+62^5\)

\({\color{maroon}{1023846957}}=2^5+15^5+20^5+53^5+57^5\)

\({\color{maroon}{1023897456}}=2^5+3^5+45^5+52^5+54^5\)

\({\color{maroon}{1024536789}}=34^5+37^5+45^5+51^5+52^5\)

\({\color{maroon}{1024539867}}=30^5+38^5+39^5+41^5+59^5\)

\({\color{maroon}{1024576389}}=16^5+24^5+45^5+47^5+57^5\)

⋮

Some pandigitals can be expressed in two ways, some of which are multigrades.

The multigrades here happen when using exponent \(1\) & \(5\) and they sum up to equal values.

\(\bbox[gold,3px,border:1px solid]{{\color{maroon}{1243879506}}\mathbf{\color{blue}{\;=\;}}5^5+24^5+47^5+55^5+55^5\mathbf{\color{blue}{\;=\;}}9^5+27^5+40^5+49^5+61^5}\)

\(5+24+47+55+55\mathbf{\color{blue}{\;=\;}}186\mathbf{\color{blue}{\;=\;}}9+27+40+49+61\)

\(\bbox[gold,3px,border:1px solid]{{\color{maroon}{5062178349}}\mathbf{\color{blue}{\;=\;}}2^5+12^5+63^5+63^5+79^5\mathbf{\color{blue}{\;=\;}}20^5+24^5+30^5+61^5+84^5}\)

\(2+12+63+63+79\mathbf{\color{blue}{\;=\;}}219\mathbf{\color{blue}{\;=\;}}20+24+30+61+84\)

\(\bbox[gold,3px,border:1px solid]{{\color{maroon}{8932546107}}\mathbf{\color{blue}{\;=\;}}4^5+43^5+61^5+67^5+92^5\mathbf{\color{blue}{\;=\;}}12^5+21^5+75^5+75^5+84^5}\)

\(4+43+61+67+92\mathbf{\color{blue}{\;=\;}}267\mathbf{\color{blue}{\;=\;}}12+21+75+75+84\)


Note that when the equations of the multigrades are not equal then the difference
of the sums of the base terms are multiples of \(30\) !
Here is an example with \(3*30=90\) with pandigital \(5986421307\) and
\(14^1+21^1+22^1+63^1+87^1=207\) minus
\(40^1+51^1+54^1+75^1+77^1=297\) which gives a difference of \(90\).
Can someone explain this observation mathematically ?

⋮

\({\color{maroon}{9875034126}}=1^5+51^5+54^5+81^5+89^5\)

\({\color{maroon}{9875064123}}=14^5+22^5+74^5+75^5+88^5\)

\({\color{maroon}{9875103426}}=1^5+1^5+14^5+76^5+94^5\)

\({\color{maroon}{9875134026}}=26^5+43^5+72^5+79^5+86^5\)

\({\color{maroon}{9875423601}}=30^5+38^5+52^5+81^5+90^5\)

\({\color{maroon}{9875642310}}=6^5+39^5+63^5+68^5+94^5\)

\({\color{maroon}{9876153024}}=4^5+14^5+16^5+76^5+94^5\)

\({\color{maroon}{9876204315}}=17^5+22^5+46^5+77^5+93^5\)

\({\color{maroon}{9876341025}}=30^5+48^5+54^5+56^5+97^5\)

\({\color{maroon}{9876531024}}=7^5+9^5+18^5+76^5+94^5\)


Secondly I would like you to start a list of solutions with pandigitals equal to five mixed signs terms.
Of course, thousands of pandigitals will pop up with a sum of five fifthpowers. Therefore I will make a connecting
list going up from the smallest found pandigital and going down from the largest found pandigital.
Special solutions with e.g. a rare beauty or an interesting twist may also be reported.


\({\color{blue}{1023456789}}=v^5+w^5+x^5+y^5+z^5\)

\({\color{maroon}{1023459768}}=(-51)^5+75^5+203^5+206^5+(-235)^5\)

\({\color{maroon}{1023568974}}=13^5+785^5+1130^5+1744^5+(-1788)^5\)

\({\color{maroon}{1023645798}}=(-1126)^5+1260^5+(-1273)^5+(-1332)^5+1439^5\)

\({\color{maroon}{1023694587}}=(-46)^5+(-210)^5+267^5+334^5+(-348)^5\)

⋮

\({\color{maroon}{4768523109}}=6061^5+10665^5+10920^5+11786^5+(-13953)^5\)

⋮

\({\color{maroon}{9876203415}}=430^5+(-713)^5+850^5+1456^5+(-1468)^5\)

\({\color{maroon}{9876204315}}=(-69)^5+(-373)^5+549^5+1436^5+(-1438)^5\)

\({\color{maroon}{9876451230}}=525^5+(-566)^5+(-588)^5+(-710)^5+769^5\)

\({\color{maroon}{9876502314}}=340^5+492^5+(-573)^5+(-617)^5+652^5\)

\({\color{maroon}{9876541320}}=-19^5+72^5+(-768)^5+(-1323)^5+1340^5\)

\({\color{maroon}{9876542301}}=(-180)^5+(-219)^5+324^5+381^5+(-405)^5\)

\({\color{blue}{9876543210}}=v^5+w^5+x^5+y^5+z^5\)



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