Happy New Year 2025
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$${\rm Published}$$ $${\rm Online}$$ $${\rm First}$$ $${\rm (1/1/2025)}$$
$${\rm Latest}$$ $${\rm Additions}$$ $${\rm (1/1/2025)}$$
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$${\small \rm \bf Introduction}$$
$${\small 2025^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}}$$
$${\footnotesize (k,1,n)→,3≦k≦5}$$
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Limits of search range
(検索範囲)
$${\small 2025^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}}$$
$${\footnotesize (k,1,n)→,3≦k≦32}$$
$${\small \rm The~notation~is:}$$
$${\small (k, m, n)}$$
$${\small \rm where:}$$
$${\small k = }$$$${\small \rm power}$$
$${\small m = }$$$${\small \rm number~of~left~terms}$$
$${\small n = }$$$${\small \rm number~of~right~terms}$$
$${\small \rm *x~means~that~we~have~to~add}$$$${\small \rm x}$$
$${\small \rm times~the~number.}$$
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Database (b=2025)
$${\small 2025^{k}=a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}}$$
$${\footnotesize (k,1,n)→,3≦k≦5}$$
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k=3
$${\footnotesize \rm (3,1,3)~2025~done~in~1~second.~7~solutions.}$$
$${\footnotesize 2025=1971+855+279}$$ $${\footnotesize \rm (divisible~by~9)}$$
$${\footnotesize (225=219+95+31)}$$
$${\footnotesize \bf 2025=1892+1137+394}$$ $${\footnotesize \rm (irreducible~solution)}$$
$${\footnotesize 2025=1850+1200+625}$$ $${\footnotesize \rm (divisible~by~25)}$$
$${\footnotesize (81=74+48+25)}$$
$${\footnotesize 2025=1800+1350+225}$$ $${\footnotesize \rm (divisible~by~225)}$$
$${\footnotesize (9=8+6+1)}$$
$${\footnotesize 2025=1782+1377+324}$$ $${\footnotesize \rm (divisible~by~81)}$$
$${\footnotesize (25=22+17+4)}$$
$${\footnotesize 2025=1782+1161+1026}$$ $${\footnotesize \rm (divisible~by~27)}$$
$${\footnotesize (75=66+43+38)}$$
$${\footnotesize 2025=1641+1572+36}$$ $${\footnotesize \rm (divisible~by~3)}$$
$${\footnotesize (675=547+524+9)}$$
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k=4
$${\footnotesize \rm (4,1,5)~2025~done~in~46~seconds.~110~solutions.}$$
2025=2021+564+402+254+178
2025=2011+762+586+266+108
2025=1983+1046+592+382+324
2025=1965+1170+420+180+60
2025=1965+1110+780+360+180
2025=1957+1194+572+298+54
2025=1957+954+918+812+646
2025=1947+1134+942+234+168
2025=1897+1244+976+828+552
2025=1857+1284+1188+594+546
2025=1793+1494+1008+826+4
2025=1791+1506+988+802+352
2025=1773+1616+576+214+196
2025=1773+1554+996+552+396
2025=1755+1620+810+270*2
2025=1755+1620+790+470+230
2025=1756+1470+1230+780+90
2025=1755+1320+1290+1110+270
2025=1729+1448+1298+894+264
2025=1727+1664+592+522+486
2025=1701+1482+1266+912+774
2025=1683+1444+1306+1104+472
2025=1665+1710+870+240*2
2025=1665+1530+1380+390+120
2025=1639+1746+734+342+194
2025=1623+1404+1344+1272+576
2025=1573+1562+1456+702+234
2025=1573+1518+1512+614+346
2025=1497+1794+954+882+186
2025=1491+1668+1256+1126+436
2025=1473+1736+1244+846+586
2025=1471+1664+1446+554+48
2025=1449+1876+376+168+56
2025=1449+1816+1112+196+144
2025=1449+1776+1164+888+144
2025=1435+1790+1230+360+230
2025=1407+1664+1392+1102+58
2025=1389+1744+1216+1126+468
2025=1381+1876+858+694+366
2025=1371+1872+846+726+678
2025=1371+1678+1372+1158+334
2025=1363+1866+1024+612+52
2025=1345+1620+1370+1380+240
2025=1317+1746+1418+812+434
2025=1307+1906+914+186+144
2025=1217+1814+1334+888+264
2025=1215+1920+980+590+230
2025=1215+1890+1170+180*2
2025=1215+1890+1080+810+540
2025=1215+1770+1440+840+390
2025=1215+1620*2+810*2
2025=1215+1570+1490+1380+260
2025=1197+1692+1368+1296+702
2025=1179+1962+480+306+108
2025=1179+1938+918+498+264
2025=1179+1892+1126+818+342
2025=1153+1628+1494+1284+754
2025=1089+1844+1222+1124+378
2025=1051+1794+1374+1132+418
2025=1023+1638+1596+1194+144
2025=997+1782+1306+1218+892
2025=959+1842+1388+884+606
2025=923+1938+1186+254+132
2025=923+1894+1232+978+222
2025=897+1926+1242+408+132
2025=897+1674+1572+1206+552
2025=873+1728+1404+1224+1044
2025=859+1998+782+446+298
2025=833+1806+1348+1242+344
2025=831+1998+744+546+288
2025=793+1614+1606+1314+94
2025=767+1914+1228+938+144
2025=767+1752+1518+1148+142
2025=741+1926+1272+606+162
2025=741+1584+1566+1302+1074
2025=729+2014+494+378+26
2025=727+2012+552+484+156
2025=727+1944+1068+988+76
2026=667+1698+1624+1062+526
2025=641+1996+922+474+168
2025=639+1674+1656+1008+702
2025=639+1632+1574+1196+1082
2025=621+1926+1176+996+312
2025=603+1998+846+594+576
2025=603+1602+1566+1404+666
2025=549+1746+1638+648+486
2025=513+2022+408+222+72
2025=513+1814+1556+486+112
2025=513+1674+1656+986+882
2025=447+1974+936+852+738
2025=385+1810+1430+1140+660
2025=385+1570+1460+1440+1170
2025=321+1906+1086+1084+956
2025=321+1848+1386+1092+414
2025=297+1692+1548+1296+468
2025=229+1902+1276+1014+362
2025=219+1898+1372+726+346
2025=151+1928+1296+644+252
2025=151+1716+1688+384+236
2025=147+1808+1572+386+124
2025=147+1746+1656+168+48
2025=83+1976+984+848+582
2025=65+1630+1610+1320+180
2025=63+1872+1296+1026+882
2025=63+1838+1404+1046+752
2025=63+1818+1404+1188+342
2025=63+1794+1488+1038+792
2025=45+1980+1080+540+90
2025=45+1960+1040+840+790
2025=45+1710+1620+1080+360
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k=5
$${\footnotesize \rm (5,1,6)~2025~done~in~44866~seconds.~12~solutions.}$$
2025=2007+1069+606+389+138+66
2025=1989+1221+700+515+279+261
2025=1958+1332+1011+468+229+177
2025=1903+1387+1258+895+690+602
2025=1880+1577+870+682+613+603
2025=1875+1485+1203+906+879+87
2025=1871+1590+825+807+749+173
2025=1865+1400+1320+1160+280+50
2025=1800+1715+785+460+165+100
2025=1725+1560+1395+1305+825+615
2025=1686+1679+1452+888+587+383
2025=1593*2+1485+1332+1107+855
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