## A^3 + B^3 + C^3 + D^3 = N^2, (A,B,C,D) in A.P.

$A^3 \; + \; B^3 \; + \; C^3 \; + \; D^3 \; = \; N^2$

A,   B,   C,   and   D   form an arithmetic progression.
Let   d   be the common difference.

d = 1 ……..   $1^3 + 2^3 + 3^3 + 4^3 = 10^2$
d = 14 ……   $39^3 + 53^3 + 67^3 + 81^3 = 1020^2$
d = 22 ……   $27^3 + 49^3 + 71^3 + 93^3 = 1140^2$
d = 23 ……   $6699^3 + 6722^3 + 6745^3 + 6768^3 = 1105098^2$
d = 29 ……   $210^3 + 239^3 + 268^3 + 297^3 = 8268^2$
d = 32 ……   $1^3 + 33^3 + 65^3 + 97^3 = 1106^2$
d = 34 ……   $9^3 + 43^3 + 77^3 + 111^3 = 1380^2$
d = 39 ……   $4^3 + 43^3 + 82^3 + 121^3 = 1550^2$
d = 79 ……   $1204^3 + 1283^3 + 1362^3 + 1441^3 = 96830^2$
d = 98 ……   $93^3 + 191^3 + 289^3 + 387^3 = 9480^2$

d = 154 …..   $1929^3 + 2083^3 + 2237^3 + 2391^3 = 202680^2$
d = 277 …..   $487^3 + 764^3 + 1041^3 + 1318^3 = 63080^2$
d = 283 …..   $4198^3 + 4481^3 + 4764^3 + 5047^3 = 632960^2$
d = 283 …..   $9417^3 + 9700^3 + 9983^3 + 10266^3 = 1955664^2$
d = 313 …..   $2304^3 + 2617^3 + 2930^3 + 3243^3 = 299022^2$
d = 347 …..   $1582^3 + 1929^3 + 2276^3 + 2623^3 = 202420^2$
d = 359 …..   $555^3 + 914^3 + 1273^3 + 1632^3 = 85698^2$
d = 382 …..   $8067^3 + 8449^3 + 8831^3 + 9213^3 = 1612080^2$
d = 387 …..   $982^3 + 1369^3 + 1756^3 + 2143^3 = 137000^2$
d = 389 …..   $858^3 + 1247^3 + 1636^3 + 2025^3 = 123504^2$
d = 453 …..   $223^3 + 676^3 + 1129^3 + 1582^3 = 75620^2$
d = 529 …..   $1488^3 + 2017^3 + 2546^3 + 3075^3 = 238914^2$
d = 563 …..   $478^3 + 1041^3 + 1604^3 + 2167^3 = 124660^2$
d = 601 …..   $6121^3 + 6722^3 + 7323^3 + 7924^3 = 1193030^2$
d = 631 …..   $1156^3 + 1787^3 + 2418^3 + 3049^3 = 223010^2$
d = 773 …..   $1122^3 + 1895^3 + 2668^3 + 3441^3 = 260676^2$
d = 816 …..   $3265^3 + 4081^3 + 4897^3 + 5713^3 = 637706^2$
d = 831 …..   $4276^3 + 5107^3 + 5938^3 + 6769^3 = 854930^2$
d = 839 …..   $795^3 + 1634^3 + 2473^3 + 3312^3 = 237318^2$
d = 863 …..   $147^3 + 1010^3 + 1873^3 + 2736^3 = 167586^2$
d = 883 …..   $9009^3 + 9892^3 + 10775^3 + 11658^3 = 2129448^2$
d = 893 …..   $714^3 + 1607^3 + 2500^3 + 3393^3 = 243312^2$
d = 976 …..   $2017^3 + 2993^3 + 3969^3 + 4945^3 = 467398^2$

d = 1280 ….   $1801^3 + 3081^3 + 4361^3 + 5641^3 = 545462^2$
d = 1307 ….   $9262^3 + 10569^3 + 11876^3 + 13183^3 = 2437460^2$
d = 1387 ….   $441^3 + 1828^3 + 3215^3 + 4602^3 = 369984^2$
d = 1583 ….   $5187^3 + 6770^3 + 8353^3 + 9936^3 = 1419006^2$
d = 1747 ….   $1593^3 + 3340^3 + 5087^3 + 6834^3 = 701508^2$
d = 1747 ….   $2902^3 + 4649^3 + 6396^3 + 8143^3 = 962560^2$
d = 1959 ….   $1684^3 + 3643^3 + 5602^3 + 7561^3 = 813130^2$
d = 2128 ….   $1849^3 + 3977^3 + 6105^3 + 8233^3 = 924562^2$
d = 2285 ….   $9546^3 + 11831^3 + 14116^3 + 16401^3 = 3122568^2$
d = 2656 ….   $1945^3 + 4601^3 + 7257^3 + 9913^3 = 1208746^2$
d = 2882 ….   $7437^3 + 10319^3 + 13201^3 + 16083^3 = 2823240^2$
d = 3472 ….   $9433^3 + 12905^3 + 16377^3 + 19849^3 = 3898862^2$
d = 3991 ….   $5236^3 + 9227^3 + 13218^3 + 17209^3 = 2887030^2$
d = 4273 ….   $5472^3 + 9745^3 + 14018^3 + 18291^3 = 3156474^2$
d = 4306 ….   $7041^3 + 11347^3 + 15653^3 + 19959^3 = 3687300^2$
d = 4443 ….   $358^3 + 4801^3 + 9244^3 + 13687^3 = 1861360^2$
d = 4464 ….   $7465^3 + 11929^3 + 16393^3 + 20857^3 = 3948658^2$
d = 4717 ….   $5527^3 + 10244^3 + 14961^3 + 19678^3 = 3494620^2$
d = 4871 ….   $9867^3 + 14738^3 + 19609^3 + 24480^3 = 5135358^2$
d = 5373 ….   $4543^3 + 9916^3 + 15289^3 + 20662^3 = 3669280^2$
d = 5424 ….   $2473^3 + 7897^3 + 13321^3 + 18745^3 = 3075374^2$
d = 5659 ….   $873^3 + 6532^3 + 12191^3 + 17850^3 = 2789016^2$
d = 6672 ….   $7153^3 + 13825^3 + 20497^3 + 27169^3 = 5628022^2$
d = 7267 ….   $2422^3 + 9689^3 + 16956^3 + 24223^3 = 4473440^2$
d = 9965 ….   $2226^3 + 12191^3 + 22156^3 + 32121^3 = 6770532^2$

math grad - Interest: Number theory
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### 3 Responses to A^3 + B^3 + C^3 + D^3 = N^2, (A,B,C,D) in A.P.

1. paul says:

This is a list If there is a solution for d, d<=100 where A <= 10000

d = 1 … 1^3 + 2^3 + 3^3 + 4^3 = 10^2
d = 2 … 12^3 + 14^3 + 16^3 + 18^3 = 120^2
d = 3 … 9^3 + 12^3 + 15^3 + 18^3 = 108^2
d = 4 … 4^3 + 8^3 + 12^3 + 16^3 = 80^2
d = 4 … 234^3 + 238^3 + 242^3 + 246^3 = 7440^2
d = 5 … 55^3 + 60^3 + 65^3 + 70^3 = 1000^2
d = 8 … 48^3 + 56^3 + 64^3 + 72^3 = 960^2
d = 9 … 9^3 + 18^3 + 27^3 + 36^3 = 270^2
d = 12 … 36^3 + 48^3 + 60^3 + 72^3 = 864^2
d = 13 … 234^3 + 247^3 + 260^3 + 273^3 = 8112^2
d = 14 … 28^3 + 42^3 + 56^3 + 70^3 = 784^2
d = 14 … 39^3 + 53^3 + 67^3 + 81^3 = 1020^2
d = 14 … 100^3 + 114^3 + 128^3 + 142^3 = 2728^2
d = 14 … 6594^3 + 6608^3 + 6622^3 + 6636^3 = 1076040^2
d = 15 … 99^3 + 114^3 + 129^3 + 144^3 = 2754^2
d = 16 … 16^3 + 32^3 + 48^3 + 64^3 = 640^2
d = 16 … 936^3 + 952^3 + 968^3 + 984^3 = 59520^2
d = 18 … 108^3 + 126^3 + 144^3 + 162^3 = 3240^2
d = 19 … 3420^3 + 3439^3 + 3458^3 + 3477^3 = 405042^2
d = 20 … 220^3 + 240^3 + 260^3 + 280^3 = 8000^2
d = 22 … 27^3 + 49^3 + 71^3 + 93^3 = 1140^2
d = 23 … 6699^3 + 6722^3 + 6745^3 + 6768^3 = 1105098^2
d = 23 … 9637^3 + 9660^3 + 9683^3 + 9706^3 = 1902284^2
d = 25 … 25^3 + 50^3 + 75^3 + 100^3 = 1250^2
d = 27 … 81^3 + 108^3 + 135^3 + 162^3 = 2916^2
d = 29 … 210^3 + 239^3 + 268^3 + 297^3 = 8268^2
d = 30 … 4^3 + 34^3 + 64^3 + 94^3 = 1064^2
d = 31 … 372^3 + 403^3 + 434^3 + 465^3 = 17298^2
d = 31 … 1395^3 + 1426^3 + 1457^3 + 1488^3 = 109554^2
d = 32 … 1^3 + 33^3 + 65^3 + 97^3 = 1106^2
d = 32 … 192^3 + 224^3 + 256^3 + 288^3 = 7680^2
d = 34 … 9^3 + 43^3 + 77^3 + 111^3 = 1380^2
d = 36 … 36^3 + 72^3 + 108^3 + 144^3 = 2160^2
d = 36 … 2106^3 + 2142^3 + 2178^3 + 2214^3 = 200880^2
d = 38 … 2508^3 + 2546^3 + 2584^3 + 2622^3 = 259920^2
d = 39 … 4^3 + 43^3 + 82^3 + 121^3 = 1550^2
d = 42 … 477^3 + 519^3 + 561^3 + 603^3 = 25380^2
d = 43 … 4558^3 + 4601^3 + 4644^3 + 4687^3 = 628660^2
d = 45 … 495^3 + 540^3 + 585^3 + 630^3 = 27000^2
d = 48 … 144^3 + 192^3 + 240^3 + 288^3 = 6912^2
d = 49 … 49^3 + 98^3 + 147^3 + 196^3 = 3430^2
d = 50 … 300^3 + 350^3 + 400^3 + 450^3 = 15000^2
d = 51 … 1071^3 + 1122^3 + 1173^3 + 1224^3 = 78030^2
d = 52 … 936^3 + 988^3 + 1040^3 + 1092^3 = 64896^2
d = 56 … 112^3 + 168^3 + 224^3 + 280^3 = 6272^2
d = 56 … 156^3 + 212^3 + 268^3 + 324^3 = 8160^2
d = 56 … 400^3 + 456^3 + 512^3 + 568^3 = 21824^2
d = 60 … 396^3 + 456^3 + 516^3 + 576^3 = 22032^2
d = 62 … 31^3 + 93^3 + 155^3 + 217^3 = 3844^2
d = 62 … 2308^3 + 2370^3 + 2432^3 + 2494^3 = 235592^2
d = 64 … 64^3 + 128^3 + 192^3 + 256^3 = 5120^2
d = 64 … 3744^3 + 3808^3 + 3872^3 + 3936^3 = 476160^2
d = 66 … 22^3 + 88^3 + 154^3 + 220^3 = 3872^2
d = 66 … 36^3 + 102^3 + 168^3 + 234^3 = 4320^2
d = 66 … 190^3 + 256^3 + 322^3 + 388^3 = 10744^2
d = 68 … 102^3 + 170^3 + 238^3 + 306^3 = 6936^2
d = 72 … 432^3 + 504^3 + 576^3 + 648^3 = 25920^2
d = 75 … 225^3 + 300^3 + 375^3 + 450^3 = 13500^2
d = 76 … 171^3 + 247^3 + 323^3 + 399^3 = 10830^2
d = 79 … 1204^3 + 1283^3 + 1362^3 + 1441^3 = 96830^2
d = 80 … 880^3 + 960^3 + 1040^3 + 1120^3 = 64000^2
d = 81 … 81^3 + 162^3 + 243^3 + 324^3 = 7290^2
d = 85 … 2941^3 + 3026^3 + 3111^3 + 3196^3 = 340442^2
d = 87 … 1972^3 + 2059^3 + 2146^3 + 2233^3 = 193430^2
d = 88 … 108^3 + 196^3 + 284^3 + 372^3 = 9120^2
d = 92 … 322^3 + 414^3 + 506^3 + 598^3 = 21160^2
d = 98 … 93^3 + 191^3 + 289^3 + 387^3 = 9480^2
d = 98 … 588^3 + 686^3 + 784^3 + 882^3 = 41160^2
d = 100 … 100^3 + 200^3 + 300^3 + 400^3 = 10000^2
d = 100 … 5850^3 + 5950^3 + 6050^3 + 6150^3 = 930000^2

Paul.

• benvitalis says:

Cool! Thanks for completing the list

2. paul says:

Here is a link to the dropbox with quite a few more