9
$\begingroup$

this is my first time posting and I hope someone can help because this question has been driving me crazy... For a bit of background I am a sophomore math major in college, I have taken math up through vector calculus and Differential equations - I came across this problem during the AMATYC Student Math League exam.

The equation $a^3 + b^2 + c^2 = 2013$ has a solution in positive integers for which $b$ is a multiple of $5$; find $a, b$, and $c$.

Finding any solution is trivial, but I am not familiar with how to find an integer solution.

Thanks for any help in advance!

$\endgroup$
7
$\begingroup$

$$ 2013 - 10^3 = 1013 $$ is prime and $$ 1013 \equiv 1 \pmod 4. $$

There is guaranteed to be an expression, and $$ 1013 = 22^2 + 23^2. $$

So $$ 10^3 + 22^2 + 23^2 = 2013. $$ The other solutions are $$ 2^3 + 22^2 + 39^2 = 2013, $$ $$ 2^3 + 18^2 + 41^2 = 2013, $$ $$ \color{magenta}{ 4^3 + 10^2 + 43^2 = 2013. } $$

Evidently that is the one they want, $(4,10,43).$ Meanwhile, there are 434 known positive integers that have no such expression, the largest being 5,042,631. However, if we do not restrict the sign of your $a,$ every integer $n$ has an expression as $$ n = a^3 + b^2 + c^2, $$ and there is an explicit recipe for finding an expression, which generally produces $a < 0.$ For example $$ 16028054332129464515^2 + 6357026807909^2 - 6357024286595^3 = 5042631 $$

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

=-=-=-=-=-=-=-=-=-=-= enter image description here =-=-=-=-=-=-=-=-=-=-=

| cite | improve this answer | |
$\endgroup$
  • 2
    $\begingroup$ If I understand correctly the only way $1013$ can be written as a sum of two squares is $22^2+23^2$ (up to permutation and signs), neither of which corresponds to $b=22,23$ being a multiple of $5$. $\endgroup$ – anon Nov 8 '13 at 5:27
  • $\begingroup$ @anon, I put in the one they want in purple. $\endgroup$ – Will Jagy Nov 8 '13 at 5:52
  • $\begingroup$ What method did you use to find the purple answer? Was it a matter of testing a bunch of values of 'a' to find it? $\endgroup$ – Jarek Hunger Nov 8 '13 at 6:03
  • 1
    $\begingroup$ @Prism, it is suspected that $x^2 + y^2 - z^5$ and $x^2 + y^2 - z^7$ and $x^2 + y^2 - z^q$ for odd prime $q$ are likewise universal, but I found no identities. Meanwhile, for odd composite exponent, we find $$ x^2 + y^2 - z^9 \neq 216 p^3 $$ for prime $p \equiv 1 \pmod 4.$ Similar behavior for $x^2 + y^2 - z^{15}$ or $x^2 + y^2 - z^{35}$ or $x^2 + y^2 - z^{91}$ as $91 = 7 \cdot 13.$ $\endgroup$ – Will Jagy Nov 9 '13 at 0:49
  • 1
    $\begingroup$ @WillJagy: Thanks for further information. That's a striking conjecture! When I am bored in bus, I shall play with $x^2+y^2-z^5$. :) $\endgroup$ – Prism Nov 9 '13 at 1:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.