(A*B*C) (A^2 + B^2 + C^2) = (D*E*F) (D^2 + E^2 + F^2)

Find six distinct positive integers such that

$A \, B \, C \, (A^2 + B^2 + C^2) \; = \; D \, E \, F \, (D^2 + E^2 + F^2)$

math grad - Interest: Number theory
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One Response to (A*B*C) (A^2 + B^2 + C^2) = (D*E*F) (D^2 + E^2 + F^2)

1. Paul says:

Here are a few with a < b < c <= 50. Format is
{{a, b, c, etc},{d, e, f, etc}}

{{1,2,18,11844},{3,6,7,11844}}
{{1,3,22,32604},{2,6,13,32604}}
{{1,4,40,258720},{2,10,22,258720}}
{{1,4,44,343728},{8,9,14,343728}}
{{1,7,30,199500},{2,6,25,199500}}
{{1,7,40,462000},{4,5,28,462000}}
{{1,7,48,790944},{3,16,22,790944}}
{{1,8,13,24336},{3,4,12,24336}}
{{1,10,12,29400},{2,4,15,29400}}
{{1,10,22,128700},{5,6,15,128700}}
{{1,10,47,1085700},{5,14,22,1085700}}
{{1,11,50,1442100},{2,19,30,1442100}}
{{1,12,14,57288},{4,7,11,57288}}
{{1,12,21,147672},{4,9,14,147672}}
{{1,12,25,231000},{2,11,20,231000}}
{{1,12,25,231000},{4,5,22,231000}}
{{2,11,20,231000},{4,5,22,231000}}
{{1,12,45,1171800},{4,14,25,1171800}}
{{1,16,24,319872},{2,4,34,319872}}
{{1,17,30,606900},{3,5,34,606900}}
{{1,18,40,1386000},{8,9,25,1386000}}
{{1,20,27,610200},{4,15,18,610200}}
{{1,22,28,781704},{7,12,18,781704}}
{{1,23,30,986700},{2,5,46,986700}}
{{1,25,26,846300},{10,13,14,846300}}
{{1,25,40,2226000},{6,8,35,2226000}}
{{1,28,30,1415400},{7,15,20,1415400}}
{{1,28,42,2997624},{2,12,49,2997624}}
{{1,28,45,3540600},{6,12,35,3540600}}
{{1,30,45,3950100},{10,19,22,3950100}}
{{1,35,50,6520500},{14,15,27,6520500}}
{{1,40,48,7497600},{2,30,44,7497600}}
{{2,4,30,220800},{5,8,16,220800}}
{{2,4,36,379008},{6,12,14,379008}}
{{2,6,10,16800},{4,5,8,16800}}
{{2,6,39,730548},{9,13,14,730548}}
{{2,6,44,1043328},{4,12,26,1043328}}
{{2,8,25,277200},{4,12,15,277200}}
{{2,9,50,2326500},{10,11,25,2326500}}
{{2,10,38,1176480},{4,15,24,1176480}}
{{2,10,41,1463700},{5,7,34,1463700}}
{{2,10,50,2604000},{7,16,25,2604000}}
{{2,12,46,2499456},{8,16,23,2499456}}
{{2,13,24,467376},{3,8,26,467376}}
{{2,14,15,178500},{5,6,17,178500}}
{{2,16,26,778752},{6,8,24,778752}}
{{2,20,21,709800},{5,6,28,709800}}
{{2,20,24,940800},{4,8,30,940800}}
{{2,20,31,1692600},{4,13,30,1692600}}
{{3,20,26,1692600},{4,13,30,1692600}}
{{2,20,44,4118400},{10,12,30,4118400}}
{{2,21,45,4668300},{15,18,19,4668300}}
{{2,24,28,1833216},{8,14,22,1833216}}
{{2,24,35,3032400},{3,14,40,3032400}}
{{2,24,42,4725504},{8,18,28,4725504}}
{{2,24,50,7392000},{4,22,40,7392000}}
{{2,24,50,7392000},{8,10,44,7392000}}
{{4,22,40,7392000},{8,10,44,7392000}}
{{2,25,41,4735500},{10,21,22,4735500}}
{{2,26,50,8268000},{13,16,30,8268000}}
{{2,29,45,7490700},{5,18,41,7490700}}
{{2,30,50,10212000},{10,16,37,10212000}}
{{2,39,50,15697500},{14,23,30,15697500}}
{{3,4,35,525000},{5,10,20,525000}}
{{3,10,34,1290300},{6,11,25,1290300}}
{{3,14,36,2269512},{4,19,27,2269512}}
{{3,15,44,4296600},{5,20,31,4296600}}
{{3,16,20,638400},{5,8,24,638400}}
{{3,24,39,5913648},{9,12,36,5913648}}
{{3,28,50,13830600},{10,20,37,13830600}}
{{3,30,36,7144200},{6,12,45,7144200}}
{{3,30,36,7144200},{7,25,28,7144200}}
{{6,12,45,7144200},{7,25,28,7144200}}
{{3,36,42,13920984},{12,21,33,13920984}}
{{4,5,20,176400},{8,9,10,176400}}
{{4,7,40,1864800},{6,10,30,1864800}}
{{4,7,45,2633400},{6,15,28,2633400}}
{{4,12,20,537600},{8,10,16,537600}}
{{4,15,32,2428800},{6,20,22,2428800}}
{{4,16,50,8870400},{8,24,30,8870400}}
{{4,17,35,3641400},{7,10,36,3641400}}
{{4,21,22,1738968},{6,11,28,1738968}}
{{4,21,40,6911520},{10,14,34,6911520}}
{{4,22,35,5313000},{11,20,23,5313000}}
{{4,23,33,4960824},{12,19,22,4960824}}
{{4,23,42,8921976},{7,12,46,8921976}}
{{4,23,50,14007000},{10,28,29,14007000}}
{{4,23,50,14007000},{12,25,29,14007000}}
{{4,27,30,5329800},{5,18,36,5329800}}
{{4,28,30,5712000},{10,12,34,5712000}}
{{4,29,44,14255472},{19,22,24,14255472}}
{{4,30,42,13507200},{5,32,36,13507200}}
{{4,30,42,13507200},{8,16,45,13507200}}
{{5,32,36,13507200},{8,16,45,13507200}}
{{4,33,34,10147368},{6,17,44,10147368}}
{{4,35,45,20575800},{12,15,46,20575800}}
{{4,45,48,37540800},{22,25,32,37540800}}
{{5,6,18,207900},{7,10,11,207900}}
{{5,6,45,2816100},{14,15,18,2816100}}
{{5,7,36,1726200},{10,12,21,1726200}}
{{5,7,44,3095400},{10,11,28,3095400}}
{{5,7,50,4504500},{13,14,25,4504500}}
{{5,12,21,768600},{7,10,20,768600}}
{{5,14,38,4428900},{9,10,35,4428900}}
{{5,18,29,3105900},{7,10,34,3105900}}
{{5,20,24,2402400},{10,14,22,2402400}}
{{5,22,24,2864400},{8,11,30,2864400}}
{{5,24,32,6240000},{6,8,50,6240000}}
{{5,26,38,10596300},{13,19,30,10596300}}
{{5,27,39,11977875},{9,13,45,11977875}}
{{5,31,42,17902500},{21,22,25,17902500}}
{{5,35,46,27096300},{21,23,30,27096300}}
{{5,36,40,21031200},{10,18,46,21031200}}
{{5,39,50,39448500},{13,34,35,39448500}}
{{6,13,35,3903900},{10,15,26,3903900}}
{{6,14,50,11474400},{7,20,40,11474400}}
{{6,18,30,4082400},{12,15,24,4082400}}
{{6,21,44,13377672},{11,19,36,13377672}}
{{6,26,39,13585572},{14,22,29,13585572}}
{{6,28,30,8668800},{9,20,32,8668800}}
{{6,32,40,20428800},{10,16,48,20428800}}
{{7,11,44,7135128},{9,14,36,7135128}}
{{7,12,48,10067904},{9,22,32,10067904}}
{{7,15,34,5105100},{11,13,30,5105100}}
{{7,21,50,21976500},{14,23,35,21976500}}
{{7,22,39,12336324},{11,13,42,12336324}}
{{7,30,36,16972200},{9,20,42,16972200}}
{{7,35,39,26706225},{13,15,49,26706225}}
{{8,10,40,5644800},{16,18,20,5644800}}
{{8,24,40,17203200},{16,20,32,17203200}}
{{8,26,35,14305200},{13,14,40,14305200}}
{{8,35,41,34095600},{9,24,50,34095600}}
{{9,10,48,10735200},{14,18,30,10735200}}
{{9,25,34,14244300},{14,17,35,14244300}}
{{9,36,38,34732152},{19,28,31,34732152}}
{{10,12,36,6652800},{14,20,22,6652800}}
{{10,14,36,8023680},{12,15,32,8023680}}
{{10,14,45,14622300},{11,15,42,14622300}}
{{10,18,28,6088320},{15,16,24,6088320}}
{{10,23,43,24507420},{14,15,46,24507420}}
{{10,23,44,25957800},{12,19,45,25957800}}
{{10,24,42,24595200},{14,20,40,24595200}}
{{10,27,46,36576900},{18,30,31,36576900}}
{{10,29,35,21984900},{15,19,38,21984900}}
{{10,30,50,52500000},{14,25,48,52500000}}
{{10,30,50,52500000},{20,25,40,52500000}}
{{10,33,36,29521800},{15,18,44,29521800}}
{{10,35,36,33024600},{14,20,45,33024600}}
{{10,40,48,76876800},{20,28,44,76876800}}
{{10,44,45,80407800},{25,31,36,80407800}}
{{11,35,42,50288700},{15,22,49,50288700}}
{{11,40,48,85008000},{20,28,46,85008000}}
{{12,15,31,7421400},{18,19,20,7421400}}
{{12,30,49,60769800},{25,26,36,60769800}}
{{13,30,45,54299700},{25,26,34,54299700}}
{{14,30,33,30284100},{19,22,35,30284100}}
{{15,28,50,73689000},{22,25,44,73689000}}

Paul.