$$ 2013 - 10^3 = 1013 $$ es primo y
$$ 1013 \equiv 1 \pmod 4. $$
No se garantiza que sea una expresión, y
$$ 1013 = 22^2 + 23^2. $$
Así
$$ 10^3 + 22^2 + 23^2 = 2013. $$
Las otras soluciones son
$$ 2^3 + 22^2 + 39^2 = 2013, $$
$$ 2^3 + 18^2 + 41^2 = 2013, $$
$$ \color{magenta}{ 4^3 + 10^2 + 43^2 = 2013. } $$
Evidently that is the one they want, $(4,10,43).$
Mientras tanto, hay 434 conocido enteros positivos que no tienen esa expresión, el más grande de 5,042,631. Sin embargo, si no restringimos el signo de su $a,$ cada entero $n$ tiene una expresión como $$ n = a^3 + b^2 + c^2, $$ and there is an explicit recipe for finding an expression, which generally produces $a < 0.$ Por ejemplo
$$ 16028054332129464515^2 + 6357026807909^2 - 6357024286595^3 = 5042631 $$
Note that, for $n \geq 0,$ we find $5042631 - n^3$ is never the sum of two squares, as (if positive) it is always divisible by some prime $q \equiv 3 \pmod 4$ but not divisible by $q^2.$
5042631 - 0^3 = 5042631 - 0 = 5042631 = 3 * 11 * 41 * 3727
5042631 - 1^3 = 5042631 - 1 = 5042630 = 2 * 5 * 47 * 10729
5042631 - 2^3 = 5042631 - 8 = 5042623 = 1109 * 4547
5042631 - 3^3 = 5042631 - 27 = 5042604 = 2^2 * 3 * 7 * 173 * 347
5042631 - 4^3 = 5042631 - 64 = 5042567 = 43 * 117269
5042631 - 5^3 = 5042631 - 125 = 5042506 = 2 * 7 * 17 * 21187
5042631 - 6^3 = 5042631 - 216 = 5042415 = 3 * 5 * 7 * 48023
5042631 - 7^3 = 5042631 - 343 = 5042288 = 2^4 * 29 * 10867
5042631 - 8^3 = 5042631 - 512 = 5042119 = 31 * 162649
5042631 - 9^3 = 5042631 - 729 = 5041902 = 2 * 3 * 31 * 27107
5042631 - 10^3 = 5042631 - 1000 = 5041631 = 7 * 19 * 37907
5042631 - 11^3 = 5042631 - 1331 = 5041300 = 2^2 * 5^2 * 11 * 4583
5042631 - 12^3 = 5042631 - 1728 = 5040903 = 3 * 7 * 240043
5042631 - 13^3 = 5042631 - 2197 = 5040434 = 2 * 7^2 * 19 * 2707
5042631 - 14^3 = 5042631 - 2744 = 5039887 = 31 * 162577
5042631 - 15^3 = 5042631 - 3375 = 5039256 = 2^3 * 3 * 19 * 43 * 257
5042631 - 16^3 = 5042631 - 4096 = 5038535 = 5 * 631 * 1597
5042631 - 17^3 = 5042631 - 4913 = 5037718 = 2 * 7 * 359837
5042631 - 18^3 = 5042631 - 5832 = 5036799 = 3 * 419 * 4007
5042631 - 19^3 = 5042631 - 6859 = 5035772 = 2^2 * 7 * 179849
5042631 - 20^3 = 5042631 - 8000 = 5034631 = 7 * 23 * 31271
5042631 - 21^3 = 5042631 - 9261 = 5033370 = 2 * 3 * 5 * 167779
5042631 - 22^3 = 5042631 - 10648 = 5031983 = 11 * 17 * 71 * 379
5042631 - 23^3 = 5042631 - 12167 = 5030464 = 2^6 * 83 * 947
5042631 - 24^3 = 5042631 - 13824 = 5028807 = 3 * 7 * 43 * 5569
5042631 - 25^3 = 5042631 - 15625 = 5027006 = 2 * 2513503
5042631 - 26^3 = 5042631 - 17576 = 5025055 = 5 * 7 * 143573
5042631 - 27^3 = 5042631 - 19683 = 5022948 = 2^2 * 3 * 7 * 59797
5042631 - 28^3 = 5042631 - 21952 = 5020679 = 1637 * 3067
5042631 - 29^3 = 5042631 - 24389 = 5018242 = 2 * 19 * 132059
5042631 - 30^3 = 5042631 - 27000 = 5015631 = 3 * 79 * 21163
5042631 - 31^3 = 5042631 - 29791 = 5012840 = 2^3 * 5 * 7 * 17903
5042631 - 32^3 = 5042631 - 32768 = 5009863 = 19 * 263677
5042631 - 33^3 = 5042631 - 35937 = 5006694 = 2 * 3 * 7 * 11 * 10837
5042631 - 34^3 = 5042631 - 39304 = 5003327 = 7 * 19 * 37619
5042631 - 35^3 = 5042631 - 42875 = 4999756 = 2^2 * 1249939
5042631 - 36^3 = 5042631 - 46656 = 4995975 = 3 * 5^2 * 29 * 2297
5042631 - 37^3 = 5042631 - 50653 = 4991978 = 2 * 107 * 23327
5042631 - 38^3 = 5042631 - 54872 = 4987759 = 7^2 * 137 * 743
5042631 - 39^3 = 5042631 - 59319 = 4983312 = 2^4 * 3 * 17 * 31 * 197
5042631 - 40^3 = 5042631 - 64000 = 4978631 = 7 * 31 * 22943
5042631 - 41^3 = 5042631 - 68921 = 4973710 = 2 * 5 * 7 * 41 * 1733
5042631 - 42^3 = 5042631 - 74088 = 4968543 = 3 * 499 * 3319
5042631 - 43^3 = 5042631 - 79507 = 4963124 = 2^2 * 23 * 73 * 739
5042631 - 44^3 = 5042631 - 85184 = 4957447 = 11 * 450677
5042631 - 45^3 = 5042631 - 91125 = 4951506 = 2 * 3 * 7 * 31 * 3803
5042631 - 46^3 = 5042631 - 97336 = 4945295 = 5 * 989059
5042631 - 47^3 = 5042631 - 103823 = 4938808 = 2^3 * 7^2 * 43 * 293
5042631 - 48^3 = 5042631 - 110592 = 4932039 = 3 * 7 * 19 * 47 * 263
5042631 - 49^3 = 5042631 - 117649 = 4924982 = 2 * 467 * 5273
5042631 - 50^3 = 5042631 - 125000 = 4917631 = 4917631
5042631 - 51^3 = 5042631 - 132651 = 4909980 = 2^2 * 3 * 5 * 19 * 59 * 73
5042631 - 52^3 = 5042631 - 140608 = 4902023 = 7 * 53 * 73 * 181
5042631 - 53^3 = 5042631 - 148877 = 4893754 = 2 * 19 * 89 * 1447
5042631 - 54^3 = 5042631 - 157464 = 4885167 = 3 * 7 * 353 * 659
5042631 - 55^3 = 5042631 - 166375 = 4876256 = 2^5 * 7 * 11 * 1979
5042631 - 56^3 = 5042631 - 175616 = 4867015 = 5 * 17 * 57259
5042631 - 57^3 = 5042631 - 185193 = 4857438 = 2 * 3 * 631 * 1283
5042631 - 58^3 = 5042631 - 195112 = 4847519 = 43 * 79 * 1427
5042631 - 59^3 = 5042631 - 205379 = 4837252 = 2^2 * 7 * 172759
5042631 - 60^3 = 5042631 - 216000 = 4826631 = 3 * 601 * 2677
5042631 - 61^3 = 5042631 - 226981 = 4815650 = 2 * 5^2 * 7 * 13759
5042631 - 62^3 = 5042631 - 238328 = 4804303 = 7^2 * 98047
5042631 - 63^3 = 5042631 - 250047 = 4792584 = 2^3 * 3 * 397 * 503
5042631 - 64^3 = 5042631 - 262144 = 4780487 = 4780487
5042631 - 65^3 = 5042631 - 274625 = 4768006 = 2 * 29 * 82207
5042631 - 66^3 = 5042631 - 287496 = 4755135 = 3 * 5 * 7 * 11 * 23 * 179
5042631 - 67^3 = 5042631 - 300763 = 4741868 = 2^2 * 19 * 43 * 1451
5042631 - 68^3 = 5042631 - 314432 = 4728199 = 7 * 675457
5042631 - 69^3 = 5042631 - 328509 = 4714122 = 2 * 3 * 7 * 112241
5042631 - 70^3 = 5042631 - 343000 = 4699631 = 19 * 31 * 79 * 101
5042631 - 71^3 = 5042631 - 357911 = 4684720 = 2^4 * 5 * 31 * 1889
5042631 - 72^3 = 5042631 - 373248 = 4669383 = 3 * 19 * 81919
5042631 - 73^3 = 5042631 - 389017 = 4653614 = 2 * 7 * 17 * 19553
5042631 - 74^3 = 5042631 - 405224 = 4637407 = 113 * 41039
5042631 - 75^3 = 5042631 - 421875 = 4620756 = 2^2 * 3 * 7 * 55009
5042631 - 76^3 = 5042631 - 438976 = 4603655 = 5 * 7 * 31 * 4243
5042631 - 77^3 = 5042631 - 456533 = 4586098 = 2 * 11 * 208459
5042631 - 78^3 = 5042631 - 474552 = 4568079 = 3 * 1522693
5042631 - 79^3 = 5042631 - 493039 = 4549592 = 2^3 * 568699
5042631 - 80^3 = 5042631 - 512000 = 4530631 = 7 * 617 * 1049
5042631 - 81^3 = 5042631 - 531441 = 4511190 = 2 * 3 * 5 * 150373
5042631 - 82^3 = 5042631 - 551368 = 4491263 = 7 * 41 * 15649
5042631 - 83^3 = 5042631 - 571787 = 4470844 = 2^2 * 7 * 159673
5042631 - 84^3 = 5042631 - 592704 = 4449927 = 3 * 1483309
5042631 - 85^3 = 5042631 - 614125 = 4428506 = 2 * 167 * 13259
5042631 - 86^3 = 5042631 - 636056 = 4406575 = 5^2 * 19 * 9277
5042631 - 87^3 = 5042631 - 658503 = 4384128 = 2^7 * 3 * 7^2 * 233
5042631 - 88^3 = 5042631 - 681472 = 4361159 = 11 * 211 * 1879
5042631 - 89^3 = 5042631 - 704969 = 4337662 = 2 * 7 * 19 * 23 * 709
5042631 - 90^3 = 5042631 - 729000 = 4313631 = 3 * 7 * 17 * 43 * 281
5042631 - 91^3 = 5042631 - 753571 = 4289060 = 2^2 * 5 * 19 * 11287
5042631 - 92^3 = 5042631 - 778688 = 4263943 = 1901 * 2243
5042631 - 93^3 = 5042631 - 804357 = 4238274 = 2 * 3 * 71 * 9949
5042631 - 94^3 = 5042631 - 830584 = 4212047 = 7 * 29 * 20749
5042631 - 95^3 = 5042631 - 857375 = 4185256 = 2^3 * 47 * 11131
5042631 - 96^3 = 5042631 - 884736 = 4157895 = 3 * 5 * 7^2 * 5657
5042631 - 97^3 = 5042631 - 912673 = 4129958 = 2 * 7 * 294997
5042631 - 98^3 = 5042631 - 941192 = 4101439 = 1523 * 2693
5042631 - 99^3 = 5042631 - 970299 = 4072332 = 2^2 * 3 * 11 * 30851
5042631 - 100^3 = 5042631 - 1000000 = 4042631 = 4042631
5042631 - 101^3 = 5042631 - 1030301 = 4012330 = 2 * 5 * 7 * 31 * 43^2
5042631 - 102^3 = 5042631 - 1061208 = 3981423 = 3 * 31^2 * 1381
5042631 - 103^3 = 5042631 - 1092727 = 3949904 = 2^4 * 7 * 35267
5042631 - 104^3 = 5042631 - 1124864 = 3917767 = 7 * 359 * 1559
5042631 - 105^3 = 5042631 - 1157625 = 3885006 = 2 * 3 * 19 * 53 * 643
5042631 - 106^3 = 5042631 - 1191016 = 3851615 = 5 * 83 * 9281
5042631 - 107^3 = 5042631 - 1225043 = 3817588 = 2^2 * 17 * 31 * 1811
5042631 - 108^3 = 5042631 - 1259712 = 3782919 = 3 * 7 * 19^2 * 499
5042631 - 109^3 = 5042631 - 1295029 = 3747602 = 2 * 79 * 23719
5042631 - 110^3 = 5042631 - 1331000 = 3711631 = 7 * 11 * 19 * 43 * 59
5042631 - 111^3 = 5042631 - 1367631 = 3675000 = 2^3 * 3 * 5^5 * 7^2
5042631 - 112^3 = 5042631 - 1404928 = 3637703 = 23 * 158161
5042631 - 113^3 = 5042631 - 1442897 = 3599734 = 2 * 1799867
5042631 - 114^3 = 5042631 - 1481544 = 3561087 = 3 * 223 * 5323
5042631 - 115^3 = 5042631 - 1520875 = 3521756 = 2^2 * 7 * 125777
5042631 - 116^3 = 5042631 - 1560896 = 3481735 = 5 * 73 * 9539
5042631 - 117^3 = 5042631 - 1601613 = 3441018 = 2 * 3 * 7 * 81929
5042631 - 118^3 = 5042631 - 1643032 = 3399599 = 7 * 485657
5042631 - 119^3 = 5042631 - 1685159 = 3357472 = 2^5 * 239 * 439
5042631 - 120^3 = 5042631 - 1728000 = 3314631 = 3 * 1104877
5042631 - 121^3 = 5042631 - 1771561 = 3271070 = 2 * 5 * 11 * 131 * 227
5042631 - 122^3 = 5042631 - 1815848 = 3226783 = 7 * 460969
5042631 - 123^3 = 5042631 - 1860867 = 3181764 = 2^2 * 3 * 29 * 41 * 223
5042631 - 124^3 = 5042631 - 1906624 = 3136007 = 7 * 17 * 19^2 * 73
5042631 - 125^3 = 5042631 - 1953125 = 3089506 = 2 * 7 * 73 * 3023
5042631 - 126^3 = 5042631 - 2000376 = 3042255 = 3 * 5 * 202817
5042631 - 127^3 = 5042631 - 2048383 = 2994248 = 2^3 * 19 * 19699
5042631 - 128^3 = 5042631 - 2097152 = 2945479 = 2945479
5042631 - 129^3 = 5042631 - 2146689 = 2895942 = 2 * 3 * 7 * 19^2 * 191
5042631 - 130^3 = 5042631 - 2197000 = 2845631 = 2845631
5042631 - 131^3 = 5042631 - 2248091 = 2794540 = 2^2 * 5 * 7 * 19961
5042631 - 132^3 = 5042631 - 2299968 = 2742663 = 3 * 7 * 11 * 31 * 383
5042631 - 133^3 = 5042631 - 2352637 = 2689994 = 2 * 31 * 43 * 1009
5042631 - 134^3 = 5042631 - 2406104 = 2636527 = 2636527
5042631 - 135^3 = 5042631 - 2460375 = 2582256 = 2^4 * 3 * 23 * 2339
5042631 - 136^3 = 5042631 - 2515456 = 2527175 = 5^2 * 7^2 * 2063
5042631 - 137^3 = 5042631 - 2571353 = 2471278 = 2 * 79 * 15641
5042631 - 138^3 = 5042631 - 2628072 = 2414559 = 3 * 7 * 31 * 3709
5042631 - 139^3 = 5042631 - 2685619 = 2357012 = 2^2 * 7 * 84179
5042631 - 140^3 = 5042631 - 2744000 = 2298631 = 2298631
5042631 - 141^3 = 5042631 - 2803221 = 2239410 = 2 * 3 * 5 * 17 * 4391
5042631 - 142^3 = 5042631 - 2863288 = 2179343 = 47 * 89 * 521
5042631 - 143^3 = 5042631 - 2924207 = 2118424 = 2^3 * 7 * 11 * 19 * 181
5042631 - 144^3 = 5042631 - 2985984 = 2056647 = 3 * 43 * 107 * 149
5042631 - 145^3 = 5042631 - 3048625 = 1994006 = 2 * 7^2 * 20347
5042631 - 146^3 = 5042631 - 3112136 = 1930495 = 5 * 7 * 19 * 2903
5042631 - 147^3 = 5042631 - 3176523 = 1866108 = 2^2 * 3 * 155509
5042631 - 148^3 = 5042631 - 3241792 = 1800839 = 19 * 94781
5042631 - 149^3 = 5042631 - 3307949 = 1734682 = 2 * 79 * 10979
5042631 - 150^3 = 5042631 - 3375000 = 1667631 = 3 * 7 * 79411
5042631 - 151^3 = 5042631 - 3442951 = 1599680 = 2^6 * 5 * 4999
5042631 - 152^3 = 5042631 - 3511808 = 1530823 = 7 * 29 * 7541
5042631 - 153^3 = 5042631 - 3581577 = 1461054 = 2 * 3 * 7 * 43 * 809
5042631 - 154^3 = 5042631 - 3652264 = 1390367 = 11 * 126397
5042631 - 155^3 = 5042631 - 3723875 = 1318756 = 2^2 * 439 * 751
5042631 - 156^3 = 5042631 - 3796416 = 1246215 = 3 * 5 * 251 * 331
5042631 - 157^3 = 5042631 - 3869893 = 1172738 = 2 * 7 * 211 * 397
5042631 - 158^3 = 5042631 - 3944312 = 1098319 = 17 * 23 * 53^2
5042631 - 159^3 = 5042631 - 4019679 = 1022952 = 2^3 * 3 * 7 * 6089
5042631 - 160^3 = 5042631 - 4096000 = 946631 = 7^2 * 19319
5042631 - 161^3 = 5042631 - 4173281 = 869350 = 2 * 5^2 * 17387
5042631 - 162^3 = 5042631 - 4251528 = 791103 = 3 * 19 * 13879
5042631 - 163^3 = 5042631 - 4330747 = 711884 = 2^2 * 31 * 5741
5042631 - 164^3 = 5042631 - 4410944 = 631687 = 7 * 31 * 41 * 71
5042631 - 165^3 = 5042631 - 4492125 = 550506 = 2 * 3 * 11 * 19 * 439
5042631 - 166^3 = 5042631 - 4574296 = 468335 = 5 * 7 * 13381
5042631 - 167^3 = 5042631 - 4657463 = 385168 = 2^4 * 7 * 19 * 181
5042631 - 168^3 = 5042631 - 4741632 = 300999 = 3 * 100333
5042631 - 169^3 = 5042631 - 4826809 = 215822 = 2 * 31 * 59^2
5042631 - 170^3 = 5042631 - 4913000 = 129631 = 129631
5042631 - 171^3 = 5042631 - 5000211 = 42420 = 2^2 * 3 * 5 * 7 * 101
5042631 - 172^3 = 5042631 - 5088448 = -45817 = -1 * 45817
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