## ``Congruent number curves'' ny2=x3-x with |n| congruent to 6 mod 8 and less than a million: Positivity of rank, and list of curves of conjectural rank 3

For which numbers n is there a rational Pythagorean triangle of area n? This problem is known to be equivalent to: for which n does the elliptic curve

E(n): n y2 = x3 - x

have a rational point other than the point at infinity and the 2-torsion points with y=0? Such n are said to be ``congruent'' because they are also known to be the numbers that occur as the common difference (``congruum'' in Latin) of a three-term arithmetic progression of squares.

We may, and henceforth do, assume that n is a positive squarefree integer. It is conjectured that if n is congruent mod 8 to 5, 6, or 7, then n is congruent. See this page, concerning the 8k+7 case, for more background on this problem and the relevant conjectures.

Theorem. If n=8k+6 and n<106 then n is a congruent number.

Together with the corresponding results in the 8k+7 and 8k+5 cases, this shows:

E(n) has positive rank for all positive integers n<106 for which the sign of the functional equation for L(E(n),s) is -1.

The proof, combining computation and theory, proceeds as described in the 8k+7 and 8k+5 pages, except that we regard E(n) as a quadratic twist of E(2) rather than of E(1), and split the 8k+6 case into two subcases according to the parity of k, or equivalently to the residue of n modulo 16. [The curve E(2) has Weierstrass equation y2=x3-4x, and is 2-isogenous with the curve y2=x3+x; these curves have conductor 64.] If n is congruent to 14 mod 16, the rational prime 2 is split in the imaginary quadratic field of discriminant -n/2, so Heegner points yield a rational point P(n) on E(n). If n is congruent to 6 mod 16, the prime 2 is inert in that quadratic field, but a ``mock Heegner'' construction applies as in the 8k+5 case, and we again get a rational point P(n). In either case we're done unless P(n) is numerically indistinguishable from a torsion point on E(n) -- which happens for 360 values of n=16m+6, and 295 values of n=16m+14. For each of these 360+295=655 values, we guess that P(n) actually is torsion, and that E(n) has analytic rank at least 3 [by Gross-Zagier in the Heegner subcase, and by the conjectural generalization of Gross-Zagier in the ``mock Heegner'' subcase]. We then expect that E(n) has a rational non-torsion point small enough to find by direct search or by 2-descent with Cremona's MWRANK. In each case we did find at least one such point. In the table at the end of this page we list the 655 n's, and for each one give the numerator and denominator of the x-coordinate of a nontrivial rational solution of ny2=x3-x. This completes the proof.

We believe that all 655 of these curves have rank exactly 3. This is because of an upper bound on the rank, obtained by 2-descent on E(n) and/or on a curve 2-isogenous with E(n). In almost all cases, we obtain an upper bound of 3; in a handful of cases the 2-descent bound is 4 but the parity conjecture reduces this to 3. In all but 30 cases, the 2-part of the Tate-Shafarevich group SHA of E(n) is trivial if E(n) has arithmetic rank 3 as expected. The 30 exceptional cases are the following 22 in the 16k+6 subcase:
 42486 = 2*3*73*97 88502 = 2*17*19*137 112326 = 2*3*97*193 202742 = 2*17*67*89 275286 = 2*3*11*43*97 295222 = 2*17*19*457 372742 = 2*17*19*577 392502 = 2*3*11*19*313 487302 = 2*3*241*337 496774 = 2*17*19*769 503206 = 2*11*89*257 563686 = 2*17*59*281 632886 = 2*3*313*337 660822 = 2*3*241*457 667318 = 2*17*19*1033 668166 = 2*3*193*577 767814 = 2*3*73*1753 774598 = 2*11*137*257 820374 = 2*3*73*1873 831878 = 2*17*43*569 875526 = 2*3*337*433 964374 = 2*3*97*1657
and the following 8 in the 16k+14 subcase:
 142222 = 2*17*47*89 287246 = 2*31*41*113 564094 = 2*17*47*353 634382 = 2*7*113*401 646366 = 2*7*137*337 679582 = 2*31*97*113 771374 = 2*23*41*409 820046 = 2*17*89*271
In each of these cases, MWRANK finds 3 independent points and confirms arithmetic rank 3 by 2-descent on an isogenous curve, but finds a subgroup of SHA isomorphic with (Z/2Z)2.

 n n, factored numerator and denominator of x 1254 2*3*11*19 19, 3 2774 2*19*73 171, 25 3502 2*17*103 128, 25 6286 2*7*449 961, 63 7230 2*3*5*241 1445, 1 7766 2*11*353 1331, 81 9430 2*5*23*41 23, 18 9654 2*3*1609 1225, 384 12166 2*7*11*79 81, 77 12270 2*3*5*409 507, 98 12534 2*3*2089 1225, 864 16206 2*3*37*73 147, 1 17958 2*3*41*73 98, 25 19046 2*89*107 98, 9 19726 2*7*1409 2809, 9 20366 2*17*599 1849, 599 20774 2*13*17*47 81, 13 21414 2*3*43*83 169, 3 22134 2*3*7*17*31 31, 3 23446 2*19*617 1225, 9 24190 2*5*41*59 4761, 41 24414 2*3*13*313 625, 1 28806 2*3*4801 2401, 2400 29294 2*97*151 3872, 97 29614 2*13*17*67 117, 17 30270 2*3*5*1009 5043, 2 32318 2*11*13*113 112225, 113 33286 2*11*17*89 299209, 9 35286 2*3*5881 8281, 3481 36366 2*3*11*19*29 1859, 3 37862 2*11*1721 2450, 729 38982 2*3*73*89 146, 121 39630 2*3*5*1321 841, 480 40406 2*89*227 13778, 13689 40710 2*3*5*23*59 2883, 8 40894 2*7*23*127 1863, 169 41230 2*5*7*19*31 76, 14 41582 2*17*1223 2448, 2 42486 2*3*73*97 243, 49 43870 2*5*41*107 107, 98 45118 2*17*1327 1352, 25 46246 2*19*1217 1369, 152 47094 2*3*47*167 841, 334 48622 2*7*23*151 10764800, 161 51302 2*113*227 908, 900 51590 2*5*7*11*67 72, 5 55510 2*5*7*13*61 63, 2 56406 2*3*7*17*79 175, 17 56630 2*5*7*809 1089, 280 56990 2*5*41*139 90, 49 57310 2*5*11*521 961, 81 58326 2*3*9721 19321, 121 60006 2*3*73*137 242, 169 60574 2*31*977 1225, 729 65198 2*7*4657 5058001, 582624 65310 2*3*5*7*311 343, 32 67438 2*7*4817 40328, 3025 67542 2*3*11257 29403, 15625 67606 2*7*11*439 4489, 439 69326 2*17*2039 2048, 9 69870 2*3*5*17*137 272, 2 70774 2*11*3217 321651, 49 71654 2*11*3257 1849, 1408 72854 2*73*499 4491, 2809 74102 2*7*67*79 578, 25 74166 2*3*47*263 216, 47 75174 2*3*11*17*67 50, 17 75454 2*31*1217 2209, 225 76958 2*7*23*239 207, 32 77046 2*3*12841 29403, 3721 77486 2*17*43*53 37553, 1089 78422 2*113*347 23762, 18225 78526 2*7*71*79 86528, 497 84134 2*23*31*59 3751, 25 84390 2*3*5*29*97 121, 24 85470 2*3*5*7*11*37 529, 11 85702 2*73*587 34322, 13225 86086 2*7*11*13*43 99, 13 86790 2*3*5*11*263 2888, 5 88206 2*3*61*241 11881, 169 88422 2*3*14737 49923, 9025 88502 2*17*19*137 323, 225 89238 2*3*107*139 139, 75 89286 2*3*23*647 1225, 69 90174 2*3*7*19*113 57, 56 91910 2*5*7*13*101 1805, 13 93126 2*3*11*17*83 5329, 17 97422 2*3*13*1249 2523, 25 98798 2*7*7057 5041, 2016 101926 2*11*41*113 338, 113 103670 2*5*7*1481 2601, 361 104006 2*7*17*19*23 3042, 17 106358 2*7*71*107 2178, 497 112326 2*3*97*193 242, 49 112486 2*11*5113 8281, 3168 114086 2*7*29*281 4176, 242 114294 2*3*43*443 486, 43 115102 2*13*19*233 8125, 729 115598 2*7*23*359 184, 175 115702 2*17*41*83 722, 25 116966 2*233*251 242, 9 119094 2*3*23*863 21025, 1176 120414 2*3*7*47*61 54, 7 120694 2*7*37*233 529, 63 122518 2*11*5569 292681, 19800 123222 2*3*11*1867 1875, 1859 123510 2*3*5*23*179 363, 5 123710 2*5*89*139 1780, 722 123846 2*3*20641 77763, 36481 124670 2*5*7*13*137 72, 65 128030 2*5*7*31*59 45, 14 128334 2*3*73*293 147, 146 129630 2*3*5*29*149 147, 2 132046 2*103*641 5408, 361 134246 2*7*43*223 567, 121 136830 2*3*5*4561 3481, 1080 138574 2*193*359 27889, 8975 142222 2*17*47*89 800, 1 142702 2*7*10193 67712, 24025 143526 2*3*19*1259 867, 392 144142 2*97*743 13225, 743 144590 2*5*19*761 1521, 1 145254 2*3*43*563 2209, 43 147934 2*17*19*229 475, 441 148774 2*73*1019 6006578, 18769 150854 2*11*6857 480491, 205209 150990 2*3*5*7*719 1445, 7 152862 2*3*73*349 1323, 73 153454 2*7*97*113 55112, 113 153510 2*3*5*7*17*43 121, 51 153798 2*3*25633 56307, 46225 154622 2*13*19*313 475, 162 155062 2*31*41*61 2511, 2450 158070 2*3*5*11*479 139445, 1014 158774 2*7*11*1031 7569, 352 159310 2*5*89*179 90, 89 160854 2*3*17*19*83 8281, 19 161446 2*89*907 3728761, 13689 161758 2*31*2609 871200, 790321 162454 2*43*1889 2337841, 23409 163590 2*3*5*7*19*41 128, 5 164886 2*3*27481 87723, 22201 165990 2*3*5*11*503 2662, 147 167078 2*139*601 3759721, 1000800 167910 2*3*5*29*193 169, 24 168062 2*17*4943 19584, 14641 169334 2*11*43*179 3179, 43 169782 2*3*28297 57963, 55225 170254 2*7*12161 6561, 5600 170430 2*3*5*13*19*23 92, 38 170526 2*3*97*293 293, 289 171390 2*3*5*29*197 10609, 29 172334 2*199*433 2048, 1849 172374 2*3*28729 24025, 4704 173614 2*7*12401 27889, 3087 175630 2*5*7*13*193 1681, 56 176470 2*5*7*2521 5041, 1 177926 2*7*71*179 1775, 1089 179158 2*7*67*191 3481, 2144 179398 2*19*4721 227529, 3800 181718 2*43*2113 781250, 36481 182006 2*11*8273 7921, 352 183702 2*3*17*1801 110450, 22201 184710 2*3*5*47*131 141, 121 187726 2*7*11*23*53 99, 7 189854 2*7*71*191 145161, 1528 190390 2*5*79*241 160, 81 191406 2*3*19*23*73 46, 27 192270 2*3*5*13*17*29 68, 10 192894 2*3*13*2473 3721, 1225 197030 2*5*17*19*61 153, 152 199310 2*5*19*1049 20691, 289 199374 2*3*7*47*101 54, 47 200974 2*17*23*257 25281, 257 202134 2*3*59*571 1083, 59 202286 2*7*14449 29241, 343 202742 2*17*67*89 5508, 1058 204126 2*3*13*2617 3146, 529 204358 2*7*11*1327 22528, 10647 204470 2*5*7*23*127 81, 46 205782 2*3*34297 34322, 25 205886 2*113*911 5197088, 152881 207694 2*113*919 968, 49 208590 2*3*5*17*409 392, 17 209094 2*3*34849 70227, 529 209686 2*59*1777 48841, 38232 211614 2*3*13*2713 7514, 625 213670 2*5*23*929 3721, 5 214214 2*7*11*13*107 99, 8 214430 2*5*41*523 605, 441 215070 2*3*5*67*107 67, 40 215326 2*23*31*151 279, 23 215710 2*5*11*37*53 2401, 1521 215798 2*11*17*577 27889, 961 216206 2*17*6359 6465608, 1168561 218454 2*3*23*1583 38809, 2400 218926 2*17*47*137 103041, 17 220094 2*7*79*199 8649, 199 220110 2*3*5*11*23*29 40, 29 221190 2*3*5*73*101 1083, 73 221254 2*11*89*113 6889, 3168 222838 2*7*11*1447 29929, 19800 223390 2*5*89*251 84681, 4361 226614 2*3*179*211 111747, 128 227958 2*3*37993 258847800363, 241462269913 228486 2*3*113*337 338, 1 228806 2*233*491 28440882, 6039593 229326 2*3*37*1033 49729, 888 231166 2*13*17*523 523, 361 231710 2*5*17*29*47 1681, 376 232526 2*7*17*977 3025, 1071 233006 2*113*1031 163592, 47089 236910 2*3*5*53*149 245, 53 239046 2*3*39841 21025, 18816 239246 2*7*23*743 1472, 14 241710 2*3*5*7*1151 1183, 32 243638 2*43*2833 5929, 3096 245494 2*131*937 1058, 121 247798 2*19*6521 3864619, 95481 250206 2*3*11*17*223 57121, 33 250518 2*3*43*971 235225, 243 251870 2*5*89*283 283, 162 251966 2*11*13*881 5733, 2209 252246 2*3*17*2473 3829849, 2941225 252374 2*257*491 243049, 3928 252726 2*3*73*577 87723, 3481 255990 2*3*5*7*23*53 160, 1 257294 2*103*1249 2601, 103 258774 2*3*17*43*59 38328481, 43 259670 2*5*23*1129 1127, 2 261654 2*3*43609 41209, 2400 261870 2*3*5*7*29*43 29, 14 269646 2*3*13*3457 2209, 1248 270470 2*5*17*37*43 7938, 17 271830 2*3*5*13*17*41 147, 17 275286 2*3*11*43*97 24843, 11 277206 2*3*47*983 109561, 983 278718 2*3*11*41*103 1849, 824 278726 2*7*43*463 3362, 121 281198 2*23*6113 422970696, 12794929 282982 2*7*17*29*41 343, 82 284694 2*3*23*2063 51529, 46 286014 2*3*73*653 507, 146 287166 2*3*11*19*229 916, 338 287246 2*31*41*113 72, 41 290366 2*47*3089 1568, 1521 291270 2*3*5*7*19*73 54, 19 291414 2*3*17*2857 4489, 1225 294630 2*3*5*7*23*61 343, 23 295118 2*41*59*61 3101121, 1661521 295222 2*17*19*457 1682, 1225 297462 2*3*11*4507 117238, 4563 297966 2*3*53*937 3481, 937 298086 2*3*49681 1424163, 1324801 300014 2*11*13*1049 5625, 4576 302158 2*17*8887 32912, 15138 305830 2*5*7*17*257 338, 257 307166 2*383*401 392, 9 307406 2*11*89*157 539, 89 310310 2*5*7*11*13*31 117, 7 314574 2*3*13*37*109 8427, 361 316526 2*7*23*983 22201, 983 320214 2*3*83*643 1369, 83 320606 2*11*13*19*59 891, 59 322766 2*71*2273 197192, 77841 324174 2*3*97*557 363, 194 326630 2*5*89*367 3481, 3125 331606 2*11*15073 761378, 22801 334510 2*5*11*3041 3481, 2601 341502 2*3*7*47*173 5000, 329 345278 2*31*5569 52398721, 27007200 346566 2*3*11*59*89 30899, 2225 348558 2*3*7*43*193 343, 43 350014 2*7*23*1087 575, 512 350670 2*3*5*11689 51483, 6962 352294 2*353*499 966289, 225 355110 2*3*5*7*19*89 54, 35 356174 2*7*13*19*103 175, 72 360326 2*67*2689 5427, 5329 361054 2*23*47*167 7225, 3384 361302 2*3*60217 96721, 36504 363630 2*3*5*17*23*31 147, 23 364206 2*3*101*601 57963, 2738 364502 2*59*3089 1638050, 2209 367478 2*43*4273 134689, 78961 370078 2*31*47*127 174375, 512 372470 2*5*7*17*313 160, 153 372742 2*17*19*577 729, 425 376414 2*17*11071 14792, 3721 376774 2*31*59*103 1194649, 271872 377846 2*7*137*197 2809, 343 379846 2*257*739 1028, 450 381254 2*19*79*127 20691, 625 381262 2*7*113*241 124609, 16641 381270 2*3*5*71*179 125, 54 383846 2*281*683 683, 441 384854 2*337*571 2738, 2401 385014 2*3*7*89*103 96, 7 385366 2*43*4481 48680128, 7529536 386670 2*3*5*12889 13467, 578 388654 2*7*17*23*71 343, 71 390286 2*13*17*883 442, 441 392502 2*3*11*19*313 294, 19 392574 2*3*7*13*719 847, 128 394270 2*5*89*443 43245, 169 394790 2*5*11*37*97 12482, 7857 395414 2*211*937 1899, 1849 396686 2*241*823 10830681, 823 397334 2*7*101*281 2527, 2 397454 2*37*41*131 2358, 676 399254 2*89*2243 1158242, 638401 399406 2*7*47*607 29929, 1400 400582 2*7*13*31*71 175, 104 400862 2*7*11*19*137 175, 99 402254 2*17*11831 21632, 9801 402670 2*5*67*601 623045, 101761 403206 2*3*17*59*67 169, 67 403990 2*5*71*569 73960, 22201 405654 2*3*17*41*97 36481, 2328 405790 2*5*7*11*17*31 125, 62 407478 2*3*113*601 3051, 1849 407526 2*3*7*31*313 338, 313 409638 2*3*67*1019 2401, 1675 412742 2*11*73*257 514, 289 418182 2*3*69697 70590962, 36869713 419566 2*7*23*1303 736, 567 421270 2*5*103*409 42849, 41405 421590 2*3*5*13*23*47 128, 13 421894 2*11*127*151 22201, 151 422294 2*19*11113 12482, 1369 425758 2*193*1103 13939200, 3508129 427854 2*3*7*61*167 175, 8 433086 2*3*19*29*131 289, 262 434294 2*7*67*463 11449, 126 435710 2*5*11*17*233 7921, 1 438126 2*3*13*41*137 5929, 312 443086 2*7*31649 37249, 5600 443894 2*11*20177 49379, 31329 444686 2*11*17*29*41 99, 17 446894 2*7*137*233 44521, 12600 448806 2*3*131*571 116162, 108241 450926 2*7*31*1039 5041, 2232 452166 2*3*11*13*17*31 496, 54 453390 2*3*5*7*17*127 169, 85 453630 2*3*5*15121 15123, 2 454062 2*3*7*19*569 625, 513 456214 2*11*89*233 305762, 4361 466134 2*3*77689 312987, 157609 468214 2*17*47*293 1175, 882 469766 2*11*131*163 361, 163 470270 2*5*31*37*41 242, 37 470758 2*113*2083 15334722, 4264225 471814 2*7*67*503 19363, 12321 471822 2*3*13*23*263 2311247, 3 473478 2*3*23*47*73 3481, 3381 477654 2*3*79609 41209, 38400 477862 2*7*11*29*107 225, 203 480326 2*7*11*3119 46475, 2809 482694 2*3*80449 227138, 146689 483446 2*17*59*241 5618, 1521 484334 2*23*10529 15129, 5929 486094 2*7*34721 57121, 12321 486174 2*3*13*23*271 13697401, 23 487302 2*3*241*337 723, 625 489062 2*7*181*193 813967, 1458 494326 2*7*17*31*67 4225, 536 495670 2*5*7*73*97 22898, 97 495942 2*3*82657 8233682, 8151025 496774 2*17*19*769 1700, 162 498470 2*5*7*7121 94249, 5445 499686 2*3*11*67*113 289, 113 500118 2*3*19*41*107 164, 50 500774 2*31*41*197 361, 197 501486 2*3*19*53*83 121, 38 501990 2*3*5*29*577 722, 145 503206 2*11*89*257 1058, 801 504710 2*5*41*1231 54760, 43681 510510 2*3*5*7*11*13*17 88, 3 511166 2*431*593 512, 81 512134 2*7*157*233 37303, 722 512582 2*7*19*41*47 450, 329 513006 2*3*13*6577 16250, 3481 513678 2*3*11*43*181 30625, 507 515526 2*3*11*73*107 1250, 73 516198 2*3*227*379 18723, 227 516206 2*199*1297 7569, 4975 516430 2*5*43*1201 8153645, 212521 517398 2*3*7*97*127 1875, 1681 519486 2*3*11*17*463 512, 49 521774 2*11*37*641 172225, 13024 522614 2*17*19*809 8281, 5472 522966 2*3*43*2027 6241, 2187 523934 2*241*1087 28912338369, 22198019407 524342 2*7*13*43*67 225, 43 533094 2*3*23*3863 183027, 5329 540014 2*193*1399 1568, 169 542366 2*11*89*277 6135529, 1260329 543638 2*67*4057 22080601, 12201049 547230 2*3*5*17*29*37 128, 17 547246 2*7*39089 33489, 5600 549430 2*5*7*47*167 167, 162 549766 2*7*107*367 1575, 107 551854 2*17*16231 36397512, 22877089 552246 2*3*92041 351122, 259081 554262 2*3*92377 2380849, 2145624 555622 2*353*787 49977648, 39690000 556654 2*7*39761 80089, 567 557574 2*3*19*67*73 3698, 121 557926 2*83*3361 717409, 418609 561470 2*5*7*13*617 2809, 2744 561966 2*3*229*409 867, 49 561990 2*3*5*11*13*131 98, 33 563686 2*17*59*281 722, 281 563766 2*3*7*31*433 867, 1 564094 2*17*47*353 200, 153 565174 2*19*107*139 10259209, 23491 568230 2*3*5*13*31*47 125, 31 574726 2*67*4289 365397938978, 58367804369 574926 2*3*11*31*281 529, 33 577182 2*3*19*61*83 3050, 1681 577374 2*3*7*59*233 289, 177 579166 2*7*41*1009 648, 361 583310 2*5*7*13*641 641, 529 583862 2*353*827 157922, 140625 585830 2*5*7*8369 74263, 1058 586534 2*13*17*1327 134784, 112225 586758 2*3*19*5147 845899, 399675 587238 2*3*97*1009 211250, 82369 588070 2*5*7*31*271 65596, 14 588126 2*3*7*11*19*67 1369, 38 590870 2*5*7*23*367 1849, 14 593446 2*7*19*23*97 1682, 161 593814 2*3*13*23*331 363, 299 594734 2*7*23*1847 31329, 18400 595406 2*7*71*599 4761, 568 597174 2*3*99529 395307, 2809 598886 2*251*1193 388129, 242968 600270 2*3*5*11*17*107 96, 11 600990 2*3*5*13*23*67 250, 49 601590 2*3*5*11*1823 11638, 2523 603174 2*3*11*13*19*37 222, 196 604830 2*3*5*20161 17161, 3000 607254 2*3*101209 103058, 1849 607614 2*3*7*17*23*37 1701, 1 607686 2*3*101281 539726449, 1161600 613830 2*3*5*7*37*79 343, 27 614510 2*5*13*29*163 393129, 90601 621606 2*3*211*491 126025, 111619 626110 2*5*17*29*127 272, 18 627830 2*5*7*8969 4489, 4480 629142 2*3*23*47*97 104882, 25 630102 2*3*11*9547 261382, 22707 630190 2*5*11*17*337 3969, 1760 631158 2*3*11*73*131 243602, 73 632510 2*5*19*3329 49419, 17161 632870 2*5*7*9041 448063, 284258 632886 2*3*313*337 946729, 96 634382 2*7*113*401 7225, 7 634486 2*19*59*283 625, 59 638302 2*7*127*359 714025, 15463 638990 2*5*11*37*157 245, 88 640894 2*31*10337 17570592, 1868689 641334 2*3*89*1201 2403, 2401 646366 2*7*137*337 200, 137 647870 2*5*17*37*103 17408, 722 648958 2*17*19087 53312, 15138 649686 2*3*19*41*139 98, 41 653606 2*281*1163 722, 441 655046 2*7*71*659 578, 81 655894 2*17*101*191 3969, 2525 656326 2*11*29833 17161, 12672 656966 2*29*47*241 16129, 6525 657438 2*3*19*73*79 1681, 294 657854 2*97*3391 3446751537, 726925279 659406 2*3*11*97*103 200, 97 660822 2*3*241*457 659883, 25 662982 2*3*47*2351 1176, 1175 664214 2*113*2939 39602432, 5916800 665854 2*7*199*239 332928, 1 666190 2*5*7*31*307 4489, 614 666222 2*3*37*3001 6627, 625 667318 2*17*19*1033 2907, 1225 668166 2*3*193*577 578, 1 668334 2*3*23*29*167 98, 69 668590 2*5*13*37*139 41357761, 7935489 669494 2*7*17*29*97 63, 34 674958 2*3*23*67*73 11132, 1350 677766 2*3*37*43*71 17161, 11907 678286 2*7*48449 42849, 5600 678774 2*3*29*47*83 492075, 32 679126 2*7*179*271 22201, 542 679582 2*31*97*113 5929, 279 681494 2*11*30977 31329, 30625 681870 2*3*5*7*17*191 361, 21 684454 2*17*41*491 316969, 34153 684886 2*73*4691 22898, 19321 686054 2*37*73*127 136161, 36703 692310 2*3*5*47*491 363, 128 694790 2*5*17*61*67 610, 529 695222 2*11*31601 40401, 22801 697886 2*7*79*631 37249, 12600 699270 2*3*5*11*13*163 98, 65 699486 2*3*73*1597 4563, 1825 707982 2*3*11*17*631 10625, 529 710214 2*3*118369 264627, 208849 712054 2*7*181*281 14641, 4525 714670 2*5*11*73*89 1369, 55 715046 2*19*31*607 625, 589 717134 2*11*37*881 25281, 18769 719054 2*7*51361 101761, 961 719854 2*23*15649 9025, 6624 720390 2*3*5*11*37*59 539, 169 724846 2*17*21319 32768, 11449 727118 2*7*167*311 2879809, 1929184 728310 2*3*5*11*2207 22472, 1805 731526 2*3*121921 232061681378, 182360775409 731814 2*3*23*5303 7225, 3381 731894 2*83*4409 110889, 109561 732990 2*3*5*53*461 363, 98 737414 2*307*1201 291901849, 243696600 741462 2*3*191*647 4775, 2187 744430 2*5*17*29*151 4232, 2873 746102 2*7*137*389 837225, 15463 749190 2*3*5*13*17*113 243, 17 752070 2*3*5*11*43*53 11638, 243 753774 2*3*7*131*137 3481, 56 755958 2*3*7*41*439 6929, 4046 756366 2*3*13*9697 21866, 7225 760494 2*3*7*19*953 1849, 57 761534 2*37*41*251 333, 169 761582 2*103*3697 36227072, 7225 762870 2*3*5*59*431 2645, 59 762926 2*17*19*1181 1665473, 1306449 764294 2*19*20113 20402, 289 764502 2*3*47*2711 1536, 1175 764790 2*3*5*13*37*53 12769, 481 766590 2*3*5*11*23*101 147, 55 767814 2*3*73*1753 4418, 841 768822 2*3*97*1321 5230369, 1048344 771374 2*23*41*409 225, 184 772294 2*23*103*163 7987, 3249 773310 2*3*5*149*173 173, 125 774598 2*11*137*257 882, 625 775806 2*3*31*43*97 6728, 3721 776742 2*3*129457 444675, 185761 780022 2*23*31*547 50562, 15625 781710 2*3*5*71*367 361, 6 783326 2*17*23039 361128977, 23039 786934 2*257*1531 10131489, 7167473 788006 2*19*89*233 1458, 233 788062 2*11*113*317 1858131, 1723969 791182 2*7*31*1823 6561, 6200 791246 2*23*103*167 6241, 1503 792374 2*11*36017 18769, 17248 795230 2*5*281*283 104329, 13769 803606 2*47*83*103 7442, 2601 805790 2*5*19*4241 79210, 1369 805966 2*7*23*2503 42527, 2527 807166 2*191*2113 1152, 961 808790 2*5*31*2609 2560, 49 810526 2*17*31*769 648, 121 812598 2*3*135433 14207869731, 232959169 815070 2*3*5*101*269 15125, 1944 815326 2*41*61*163 163, 81 816486 2*3*11*89*139 667489, 11 818062 2*7*71*823 931225, 3703 820046 2*17*89*271 13448, 169 820374 2*3*73*1873 1849, 24 822822 2*3*7*11*13*137 625, 77 824702 2*29*59*241 43231424, 1498176 825654 2*3*23*31*193 2976, 1849 826574 2*7*17*23*151 5625, 151 830830 2*5*7*11*13*83 567, 512 831878 2*17*43*569 16129, 5121 832174 2*7*59441 75625, 16184 837718 2*7*53*1129 3345241, 2671200 838558 2*7*89*673 648, 25 840198 2*3*233*601 211250, 208849 841654 2*11*67*571 603, 539 841854 2*3*13*43*251 147, 104 845326 2*71*5953 394272, 326041 849030 2*3*5*7*13*311 2187, 10 851214 2*3*7*13*1559 1515361, 13 855998 2*11*13*41*73 584, 441 856214 2*107*4001 284258, 88209 862422 2*3*11*73*179 24649, 24576 866598 2*3*97*1489 396050, 37249 868454 2*67*6481 17725058, 17447329 872822 2*19*103*223 49419, 3481 872942 2*7*23*2711 54289, 8064 873686 2*11*151*263 79475, 49 874230 2*3*5*7*23*181 125, 56 875526 2*3*337*433 722, 289 875806 2*17*25759 773229249, 211017728 875886 2*3*11*23*577 1127, 27 876854 2*11*39857 125939, 33489 878974 2*31*14177 42025, 13671 879238 2*499*881 16136289, 399200 881790 2*3*5*7*13*17*19 189, 19 883014 2*3*11*17*787 1331, 243 883286 2*17*83*313 29963, 19321 887718 2*3*13*19*599 637, 38 888294 2*3*11*43*313 13467, 8 889878 2*3*11*97*139 13475, 8 889918 2*449*991 11746954228689, 10748087907200 893878 2*107*4177 42468300, 10355524 893926 2*11*179*227 729, 179 896790 2*3*5*167*179 1849, 179 899854 2*71*6337 11449, 1225 905334 2*3*150889 75625, 75264 907878 2*3*337*449 266450, 187489 911870 2*5*67*1361 3674889, 51529 912646 2*7*19*47*73 1058, 329 914686 2*31*14753 8162449, 358112 918030 2*3*5*71*431 1369, 355 918294 2*3*13*61*193 21218, 193 922046 2*17*47*577 27848, 729 924766 2*17*59*461 1003, 841 928814 2*41*47*241 2048, 121 929190 2*3*5*47*659 2401, 235 929230 2*5*43*2161 13927805, 1125721 929390 2*5*7*11*17*71 99, 71 938126 2*7*113*593 2272712, 1803649 939054 2*3*53*2953 4225, 1681 942214 2*7*13*31*167 334, 100 942942 2*3*7*11*13*157 150, 7 943014 2*3*73*2153 3650, 2809 949630 2*5*11*89*97 979, 961 949670 2*5*23*4129 3887, 242 953558 2*59*8081 13364450, 11029041 958126 2*13*43*857 9801, 857 958334 2*13*29*31*41 946688, 41 958398 2*3*7*19*1201 9633, 9583 960070 2*5*19*31*163 931, 279 960326 2*251*1913 4402365857, 4380510232 960510 2*3*5*101*317 1369, 101 961494 2*3*191*839 1217307, 896809 961598 2*11*109*401 674081, 194481 961894 2*17*19*1489 101251, 1 964374 2*3*97*1657 83521, 2328 964830 2*3*5*29*1109 52441, 11881 969006 2*3*29*5569 8508889, 852600 973390 2*5*11*8849 107811, 69169 973958 2*23*31*683 729, 46 975414 2*3*11*14779 28227, 1331 986590 2*5*11*8969 53290, 45369 987734 2*11*17*19*139 1377, 152 990366 2*3*13*12697 11449, 1248 992670 2*3*5*7*29*163 243, 163 993678 2*3*7*59*401 625, 177 996190 2*5*13*79*97 841, 711 996598 2*107*4657 185761, 47089 996710 2*5*11*13*17*41 99, 65 997094 2*7*67*1063 80143, 41875 998070 2*3*5*17*19*103 3211, 121