## Equal sums and sums of six 5-th powers

$A + B + C + D + E + F \; = \; G + H + I + J + K + L$

$A^5 + B^5 + C^5 + D^5 + E^5 + F^5 \; = \; G^5 + H^5 + I^5 + J^5 + K^5 + L^5$

with   A = 1   and   B,   C,   D,   E,   F   <   200

I arranged the results to produce an arithmetic progression (A.P.) between the successive equal sums:

54, 72, 90, 108, 126, 144, 162, … are in A.P. with 18 as the common difference.

$1+4+6+12+14+17 \; = \; 2+2+9+9+16+16 \; = \; 54$
$1^5+4^5+6^5+12^5+14^5+17^5 = 2^5+2^5+9^5+9^5+16^5+16^5 = 2215314$

$1+6+7+17+18+23 \; = \; 2+3+11+13+21+22 \; = \; 72$
$1^5+6^5+7^5+17^5+18^5+23^5 = 2^5+3^5+11^5+13^5+21^5+22^5 = 9770352$

$1+8+8+22+22+29 \; = \; 2+4+13+17+26+28 \; = \; 90$
$1^5+8^5+8^5+22^5+22^5+29^5 = 2^5+4^5+13^5+17^5+26^5+28^5 = 30883950$

$1+9+10+26+27+35 \; = \; 2+5+15+21+31+34 \; = \; 108$
$1^5+9^5+10^5+26^5+27^5+35^5 = 2^5+5^5+15^5+21^5+31^5+34^5 = 78911208$

$1+10+12+30+32+41 \; = \; 2+6+17+25+36+40 \; = \; 126$
$1^5+10^5+12^5+30^5+32^5+41^5 = 2^5+6^5+17^5+25^5+36^5+40^5 = 174059466$

$1+11+14+34+37+47 \; = \; 2+7+19+29+41+46 \; = \; 144$
$1^5+11^5+14^5+34^5+37^5+47^5 = 2^5+7^5+19^5+29^5+41^5+46^5 = 344823264$

$1+12+16+38+42+53 \; = \; 2+8+21+33+46+52 \; = \; 162$
$1^5+12^5+16^5+38^5+42^5+53^5 = 2^5+8^5+21^5+33^5+46^5+52^5 = 629419302$

$1+13+18+42+47+59 \; = \; 2+9+23+37+51+58 \; = \; 180$
$1^5+13^5+18^5+42^5+47^5+59^5 = 2^5+9^5+23^5+37^5+51^5+58^5 = 1077221400$

$1+14+20+46+52+65 \; = \; 2+10+25+41+56+64 \; = \; 198$
$1^5+14^5+20^5+46^5+52^5+65^5 = 2^5+10^5+25^5+41^5+56^5+64^5 = 1750195458$

$1+15+22+50+57+71 \; = \; 2+11+27+45+61+70 \; = \; 216$
$1^5+15^5+22^5+50^5+57^5+71^5 \; = \; 2^5+11^5+27^5+45^5+61^5+70^5 \; = \; 2724334416$

$1+16+24+54+62+77 \; = \; 2+12+29+49+66+76 \; = \; 234$
$1^5+16^5+24^5+54^5+62^5+77^5 \; = \; 2^5+12^5+29^5+49^5+66^5+76^5 \; = \; 4091093214$

$1+17+26+58+67+83 \; = \; 2+13+31+53+71+82 \; = \; 252$
$1^5+17^5+26^5+58^5+67^5+83^5 \; = \; 2^5+13^5+31^5+53^5+71^5+82^5 \; = \; 5958823752$

$1+18+28+62+72+89 \; = \; 2+14+33+57+76+88 \; = \; 270$
$1^5+18^5+28^5+62^5+72^5+89^5 \; = \; 2^5+14^5+33^5+57^5+76^5+88^5 \; = \; 8454209850$

$1+19+30+66+77+95 \; = \; 2+15+35+61+81+94 \; = \; 288$
$1^5+19^5+30^5+66^5+77^5+95^5 \; = \; 2^5+15^5+35^5+61^5+81^5+94^5 \; = \; 11723702208$

$1+20+32+70+82+101 \; = \; 2+16+37+65+86+100 \; = \; 306$
$1^5+20^5+32^5+70^5+82^5+101^5 \; = \; 2^5+16^5+37^5+65^5+86^5+100^5$
$= \; 15934953366$

$1+21+34+74+87+107 \; = \; 2+17+39+69+91+106 \; = \; 324$
$1^5+21^5+34^5+74^5+87^5+107^5 \; = \; 2^5+17^5+39^5+69^5+91^5+106^5$
$= \; 21278252664$

$1+22+36+78+92+113 \; = \; 2+18+41+73+96+112 \; = \; 342$
$1^5+22^5+36^5+78^5+92^5+113^5 \; = \; 2^5+18^5+41^5+73^5+96^5+112^5$
$= \; 27967961202$

$1+23+38+82+97+119 \; = \; 2+19+43+77+101+118 \; = \; 360$
$1^5+23^5+38^5+82^5+97^5+119^5 \; = \; 2^5+19^5+43^5+77^5+101^5+118^5$
$= \; 36243946800$

$1+24+40+86+102+125 \; = \; 2+20+45+81+106+124 \; = \; 378$
$1^5+24^5+40^5+86^5+102^5+125^5 \; = \; 2^5+20^5+45^5+81^5+106^5+124^5$
$= \; 46373018958$

$1+25+42+90+107+131 \; = \; 2+21+47+85+111+130 \; = \; 396$
$1^5+25^5+42^5+90^5+107^5+131^5 \; = \; 2^5+21^5+47^5+85^5+111^5+130^5$
$= \; 58650363816$

$1+26+44+94+112+137 \; = \; 2+22+49+89+116+136 \; = \; 414$
$1^5+26^5+44^5+94^5+112^5+137^5 \; = \; 2^5+22^5+49^5+89^5+116^5+136^5$
$= \; 73400979114$

$1+27+46+98+117+143 \; = \; 2+23+51+93+121+142 \; = \; 432$
$1^5+27^5+46^5+98^5+117^5+143^5 \; = \; 2^5+23^5+51^5+93^5+121^5+142^5$
$= \; 90981109152$

$1+28+48+102+122+149 \; = \; 2+24+53+97+126+148 \; = \; 450$
$1^5+28^5+48^5+102^5+122^5+149^5 \; = \; 2^5+24^5+53^5+97^5+126^5+148^5$
$= \; 111779679750$

$1+29+50+106+127+155 \; = \; 2+25+55+101+131+154 \; = \; 468$
$1^5+29^5+50^5+106^5+127^5+155^5 \; = \; 2^5+25^5+55^5+101^5+131^5+154^5$
$= \; 136219733208$

$1+30+52+110+132+161 \; = \; 2+26+57+105+136+160 \; = \; 486$
$1^5+30^5+52^5+110^5+132^5+161^5 \; = \; 2^5+26^5+57^5+105^5+136^5+160^5$
$= \; 164759863266$

$1+31+54+114+137+167 \; = \; 2+27+59+109+141+166 \; = \; 504$
$1^5+31^5+54^5+114^5+137^5+167^5 \; = \; 2^5+27^5+59^5+109^5+141^5+166^5$
$= \; 197895650064$

$1+32+56+118+142+173 \; = \; 2+28+61+113+146+172 \; = \; 522$
$1^5+32^5+56^5+118^5+142^5+173^5 \; = \; 2^5+28^5+61^5+113^5+146^5+172^5$
$= \; 236161095102$

$1+33+58+122+147+179 \; = \; 2+29+63+117+151+178 \; = \; 540$
$1^5+33^5+58^5+122^5+147^5+179^5 \; = \; 2^5+29^5+63^5+117^5+151^5+178^5$
$= \; 280130056200$

$1+34+60+126+152+185 \; = \; 2+30+65+121+156+184 \; = \; 558$
$1^5+34^5+60^5+126^5+152^5+185^5 \; = \; 2^5+30^5+65^5+121^5+156^5+184^5$
$= \; 330417682458$

$1+35+62+130+157+191 \; = \; 2+31+67+125+161+190 \; = \; 576$
$1^5+35^5+62^5+130^5+157^5+191^5 \; = \; 2^5+31^5+67^5+125^5+161^5+190^5$
$= \; 387681849216$

$1+36+64+134+162+197 \; = \; 2+32+69+129+166+196 \; = \; 594$
$1^5+36^5+64^5+134^5+162^5+197^5 \; = \; 2^5+32^5+69^5+129^5+166^5+196^5$
$= \; 452624593014$