Integers n for which n^2 – 1 is a product of 3 different primes

 
 

Find positive integers   n   for which   n^2 - 1   is a product of three different primes

 
 

For example,

n = 14, 16, 20, 22, and 32

14^2 \; - \; 1 \; = \; 3 \; \times \; 5 \; \times \; 13
16^2 \; - \; 1 \; = \; 3 \; \times \; 5 \; \times \; 17
20^2 \; - \; 1 \; = \; 3 \; \times \; 7 \; \times \; 19
22^2 \; - \; 1 \; = \; 3 \; \times \; 7 \; \times \; 23
32^2 \; - \; 1 \; = \; 3 \; \times \; 11 \; \times \; 31

 
 
Find other examples.

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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4 Responses to Integers n for which n^2 – 1 is a product of 3 different primes

  1. paul says:

    Here are a few

    11^2 – 1 = 120 = 2 x 3 x 5
    13^2 – 1 = 168 = 2 x 3 x 7
    14^2 – 1 = 195 = 3 x 5 x 13
    16^2 – 1 = 255 = 3 x 5 x 17
    19^2 – 1 = 360 = 2 x 3 x 5
    20^2 – 1 = 399 = 3 x 7 x 19
    21^2 – 1 = 440 = 2 x 5 x 11
    22^2 – 1 = 483 = 3 x 7 x 23
    23^2 – 1 = 528 = 2 x 3 x 11
    25^2 – 1 = 624 = 2 x 3 x 13
    27^2 – 1 = 728 = 2 x 7 x 13
    31^2 – 1 = 960 = 2 x 3 x 5
    32^2 – 1 = 1023 = 3 x 11 x 31
    35^2 – 1 = 1224 = 2 x 3 x 17
    36^2 – 1 = 1295 = 5 x 7 x 37
    37^2 – 1 = 1368 = 2 x 3 x 19
    38^2 – 1 = 1443 = 3 x 13 x 37
    39^2 – 1 = 1520 = 2 x 5 x 19
    40^2 – 1 = 1599 = 3 x 13 x 41
    44^2 – 1 = 1935 = 3 x 5 x 43
    45^2 – 1 = 2024 = 2 x 11 x 23
    46^2 – 1 = 2115 = 3 x 5 x 47
    47^2 – 1 = 2208 = 2 x 3 x 23
    49^2 – 1 = 2400 = 2 x 3 x 5
    50^2 – 1 = 2499 = 3 x 7 x 17
    51^2 – 1 = 2600 = 2 x 5 x 13
    52^2 – 1 = 2703 = 3 x 17 x 53
    53^2 – 1 = 2808 = 2 x 3 x 13
    54^2 – 1 = 2915 = 5 x 11 x 53
    55^2 – 1 = 3024 = 2 x 3 x 7
    57^2 – 1 = 3248 = 2 x 7 x 29
    58^2 – 1 = 3363 = 3 x 19 x 59
    62^2 – 1 = 3843 = 3 x 7 x 61
    65^2 – 1 = 4224 = 2 x 3 x 11
    66^2 – 1 = 4355 = 5 x 13 x 67
    68^2 – 1 = 4623 = 3 x 23 x 67
    70^2 – 1 = 4899 = 3 x 23 x 71
    73^2 – 1 = 5328 = 2 x 3 x 37
    74^2 – 1 = 5475 = 3 x 5 x 73
    75^2 – 1 = 5624 = 2 x 19 x 37
    78^2 – 1 = 6083 = 7 x 11 x 79
    81^2 – 1 = 6560 = 2 x 5 x 41
    84^2 – 1 = 7055 = 5 x 17 x 83
    87^2 – 1 = 7568 = 2 x 11 x 43
    88^2 – 1 = 7743 = 3 x 29 x 89
    90^2 – 1 = 8099 = 7 x 13 x 89
    93^2 – 1 = 8648 = 2 x 23 x 47
    95^2 – 1 = 9024 = 2 x 3 x 47
    96^2 – 1 = 9215 = 5 x 19 x 97
    97^2 – 1 = 9408 = 2 x 3 x 7
    98^2 – 1 = 9603 = 3 x 11 x 97
    99^2 – 1 = 9800 = 2 x 5 x 7
    100^2 – 1 = 9999 = 3 x 11 x 101

    Paul.

  2. K.D. BAJPAI says:

    [ 38^2 – 1 = 3 * 13 * 37 ]
    [ 40^2 – 1 = 3 * 13 * 41 ]

    [ 52^2 – 1 = 3 * 17 * 53 ]

    [ 58^2 – 1 = 3 * 19 * 59 ]

    [ 68^2 – 1 = 3 * 23 * 67 ]
    [ 70^2 – 1 = 3 * 23 * 71 ]

  3. K.D. BAJPAI says:

    A list is posted that is in ascending order of the values of n^2 – 1, just to avoid any duplication or omission of terms.
    Integers n for which n^2 – 1 is product of three distinct primes:
    14 ^2-1 = 195 = {3 x 5 x 13 }
    16 ^2-1 = 255 = {3 x 5 x 17 }
    20 ^2-1 = 399 = {3 x 7 x 19 }
    22 ^2-1 = 483 = {3 x 7 x 23 }
    32 ^2-1 = 1023 = {3 x 11 x 31 }
    36 ^2-1 = 1295 = {5 x 7 x 37 }
    38 ^2-1 = 1443 = {3 x 13 x 37 }
    40 ^2-1 = 1599 = {3 x 13 x 41 }
    52 ^2-1 = 2703 = {3 x 17 x 53 }
    54 ^2-1 = 2915 = {5 x 11 x 53 }
    58 ^2-1 = 3363 = {3 x 19 x 59 }
    66 ^2-1 = 4355 = {5 x 13 x 67 }
    68 ^2-1 = 4623 = {3 x 23 x 67 }
    70 ^2-1 = 4899 = {3 x 23 x 71 }
    78 ^2-1 = 6083 = {7 x 11 x 79 }
    84 ^2-1 = 7055 = {5 x 17 x 83 }
    88 ^2-1 = 7743 = {3 x 29 x 89 }
    90 ^2-1 = 8099 = {7 x 13 x 89 }
    96 ^2-1 = 9215 = {5 x 19 x 97 }
    110 ^2-1 = 12099 = {3 x 37 x 109 }
    112 ^2-1 = 12543 = {3 x 37 x 113 }
    114 ^2-1 = 12995 = {5 x 23 x 113 }
    128 ^2-1 = 16383 = {3 x 43 x 127 }
    130 ^2-1 = 16899 = {3 x 43 x 131 }
    132 ^2-1 = 17423 = {7 x 19 x 131 }
    140 ^2-1 = 19599 = {3 x 47 x 139 }
    156 ^2-1 = 24335 = {5 x 31 x 157 }
    158 ^2-1 = 24963 = {3 x 53 x 157 }
    162 ^2-1 = 26243 = {7 x 23 x 163 }
    178 ^2-1 = 31683 = {3 x 59 x 179 }
    182 ^2-1 = 33123 = {3 x 61 x 181 }
    200 ^2-1 = 39999 = {3 x 67 x 199 }
    210 ^2-1 = 44099 = {11 x 19 x 211 }
    212 ^2-1 = 44943 = {3 x 71 x 211 }
    222 ^2-1 = 49283 = {13 x 17 x 223 }
    234 ^2-1 = 54755 = {5 x 47 x 233 }
    238 ^2-1 = 56643 = {3 x 79 x 239 }
    250 ^2-1 = 62499 = {3 x 83 x 251 }
    252 ^2-1 = 63503 = {11 x 23 x 251 }
    258 ^2-1 = 66563 = {7 x 37 x 257 }
    264 ^2-1 = 69695 = {5 x 53 x 263 }
    268 ^2-1 = 71823 = {3 x 89 x 269 }
    292 ^2-1 = 85263 = {3 x 97 x 293 }
    294 ^2-1 = 86435 = {5 x 59 x 293 }
    306 ^2-1 = 93635 = {5 x 61 x 307 }
    308 ^2-1 = 94863 = {3 x 103 x 307 }
    310 ^2-1 = 96099 = {3 x 103 x 311 }
    318 ^2-1 = 101123 = {11 x 29 x 317 }
    330 ^2-1 = 108899 = {7 x 47 x 331 }
    336 ^2-1 = 112895 = {5 x 67 x 337 }
    338 ^2-1 = 114243 = {3 x 113 x 337 }
    354 ^2-1 = 125315 = {5 x 71 x 353 }
    366 ^2-1 = 133955 = {5 x 73 x 367 }
    372 ^2-1 = 138383 = {7 x 53 x 373 }
    378 ^2-1 = 142883 = {13 x 29 x 379 }
    380 ^2-1 = 144399 = {3 x 127 x 379 }
    382 ^2-1 = 145923 = {3 x 127 x 383 }
    390 ^2-1 = 152099 = {17 x 23 x 389 }
    396 ^2-1 = 156815 = {5 x 79 x 397 }
    402 ^2-1 = 161603 = {13 x 31 x 401 }
    408 ^2-1 = 166463 = {11 x 37 x 409 }
    410 ^2-1 = 168099 = {3 x 137 x 409 }
    418 ^2-1 = 174723 = {3 x 139 x 419 }
    438 ^2-1 = 191843 = {19 x 23 x 439 }
    444 ^2-1 = 197135 = {5 x 89 x 443 }
    448 ^2-1 = 200703 = {3 x 149 x 449 }
    450 ^2-1 = 202499 = {11 x 41 x 449 }
    468 ^2-1 = 219023 = {7 x 67 x 467 }
    480 ^2-1 = 230399 = {13 x 37 x 479 }
    486 ^2-1 = 236195 = {5 x 97 x 487 }
    488 ^2-1 = 238143 = {3 x 163 x 487 }
    490 ^2-1 = 240099 = {3 x 163 x 491 }
    492 ^2-1 = 242063 = {17 x 29 x 491 }
    498 ^2-1 = 248003 = {7 x 71 x 499 }
    500 ^2-1 = 249999 = {3 x 167 x 499 }
    502 ^2-1 = 252003 = {3 x 167 x 503 }
    504 ^2-1 = 254015 = {5 x 101 x 503 }
    510 ^2-1 = 260099 = {7 x 73 x 509 }
    520 ^2-1 = 270399 = {3 x 173 x 521 }
    542 ^2-1 = 293763 = {3 x 181 x 541 }
    546 ^2-1 = 298115 = {5 x 109 x 547 }
    558 ^2-1 = 311363 = {13 x 43 x 557 }
    564 ^2-1 = 318095 = {5 x 113 x 563 }
    572 ^2-1 = 327183 = {3 x 191 x 571 }
    578 ^2-1 = 334083 = {3 x 193 x 577 }
    588 ^2-1 = 345743 = {19 x 31 x 587 }
    592 ^2-1 = 350463 = {3 x 197 x 593 }
    598 ^2-1 = 357603 = {3 x 199 x 599 }
    612 ^2-1 = 374543 = {13 x 47 x 613 }
    630 ^2-1 = 396899 = {17 x 37 x 631 }
    632 ^2-1 = 399423 = {3 x 211 x 631 }
    648 ^2-1 = 419903 = {11 x 59 x 647 }
    654 ^2-1 = 427715 = {5 x 131 x 653 }
    672 ^2-1 = 451583 = {11 x 61 x 673 }
    678 ^2-1 = 459683 = {7 x 97 x 677 }
    682 ^2-1 = 465123 = {3 x 227 x 683 }
    684 ^2-1 = 467855 = {5 x 137 x 683 }
    690 ^2-1 = 476099 = {13 x 53 x 691 }
    700 ^2-1 = 489999 = {3 x 233 x 701 }
    702 ^2-1 = 492803 = {19 x 37 x 701 }
    708 ^2-1 = 501263 = {7 x 101 x 709 }
    718 ^2-1 = 515523 = {3 x 239 x 719 }
    720 ^2-1 = 518399 = {7 x 103 x 719 }
    732 ^2-1 = 535823 = {17 x 43 x 733 }
    738 ^2-1 = 544643 = {11 x 67 x 739 }
    744 ^2-1 = 553535 = {5 x 149 x 743 }
    750 ^2-1 = 562499 = {7 x 107 x 751 }
    752 ^2-1 = 565503 = {3 x 251 x 751 }
    756 ^2-1 = 571535 = {5 x 151 x 757 }
    762 ^2-1 = 580643 = {7 x 109 x 761 }
    768 ^2-1 = 589823 = {13 x 59 x 769 }
    770 ^2-1 = 592899 = {3 x 257 x 769 }
    772 ^2-1 = 595983 = {3 x 257 x 773 }
    786 ^2-1 = 617795 = {5 x 157 x 787 }
    788 ^2-1 = 620943 = {3 x 263 x 787 }
    798 ^2-1 = 636803 = {17 x 47 x 797 }
    808 ^2-1 = 652863 = {3 x 269 x 809 }
    812 ^2-1 = 659343 = {3 x 271 x 811 }
    830 ^2-1 = 688899 = {3 x 277 x 829 }
    852 ^2-1 = 725903 = {23 x 37 x 853 }
    864 ^2-1 = 746495 = {5 x 173 x 863 }
    878 ^2-1 = 770883 = {3 x 293 x 877 }
    880 ^2-1 = 774399 = {3 x 293 x 881 }
    888 ^2-1 = 788543 = {7 x 127 x 887 }
    906 ^2-1 = 820835 = {5 x 181 x 907 }
    912 ^2-1 = 831743 = {11 x 83 x 911 }
    918 ^2-1 = 842723 = {7 x 131 x 919 }
    920 ^2-1 = 846399 = {3 x 307 x 919 }
    938 ^2-1 = 879843 = {3 x 313 x 937 }
    940 ^2-1 = 883599 = {3 x 313 x 941 }
    942 ^2-1 = 887363 = {23 x 41 x 941 }
    948 ^2-1 = 898703 = {13 x 73 x 947 }
    952 ^2-1 = 906303 = {3 x 317 x 953 }
    954 ^2-1 = 910115 = {5 x 191 x 953 }
    966 ^2-1 = 933155 = {5 x 193 x 967 }
    972 ^2-1 = 944783 = {7 x 139 x 971 }
    978 ^2-1 = 956483 = {11 x 89 x 977 }
    984 ^2-1 = 968255 = {5 x 197 x 983 }
    990 ^2-1 = 980099 = {23 x 43 x 991 }
    992 ^2-1 = 984063 = {3 x 331 x 991 }
    996 ^2-1 = 992015 = {5 x 199 x 997 }
    1008 ^2-1 = 1016063 = {19 x 53 x 1009 }
    1010 ^2-1 = 1020099 = {3 x 337 x 1009 }
    1012 ^2-1 = 1024143 = {3 x 337 x 1013 }
    1038 ^2-1 = 1077443 = {17 x 61 x 1039 }
    1040 ^2-1 = 1081599 = {3 x 347 x 1039 }
    1048 ^2-1 = 1098303 = {3 x 349 x 1049 }
    1060 ^2-1 = 1123599 = {3 x 353 x 1061 }
    1068 ^2-1 = 1140623 = {11 x 97 x 1069 }
    1098 ^2-1 = 1205603 = {7 x 157 x 1097 }
    1102 ^2-1 = 1214403 = {3 x 367 x 1103 }
    1110 ^2-1 = 1232099 = {11 x 101 x 1109 }
    1116 ^2-1 = 1245455 = {5 x 223 x 1117 }
    1118 ^2-1 = 1249923 = {3 x 373 x 1117 }
    1122 ^2-1 = 1258883 = {19 x 59 x 1123 }
    1150 ^2-1 = 1322499 = {3 x 383 x 1151 }
    1164 ^2-1 = 1354895 = {5 x 233 x 1163 }
    1170 ^2-1 = 1368899 = {7 x 167 x 1171 }
    1188 ^2-1 = 1411343 = {29 x 41 x 1187 }
    1192 ^2-1 = 1420863 = {3 x 397 x 1193 }
    1194 ^2-1 = 1425635 = {5 x 239 x 1193 }
    1200 ^2-1 = 1439999 = {11 x 109 x 1201 }
    1202 ^2-1 = 1444803 = {3 x 401 x 1201 }
    1212 ^2-1 = 1468943 = {7 x 173 x 1213 }
    1218 ^2-1 = 1483523 = {23 x 53 x 1217 }
    1228 ^2-1 = 1507983 = {3 x 409 x 1229 }
    1248 ^2-1 = 1557503 = {29 x 43 x 1249 }
    1258 ^2-1 = 1582563 = {3 x 419 x 1259 }
    1260 ^2-1 = 1587599 = {13 x 97 x 1259 }
    1284 ^2-1 = 1648655 = {5 x 257 x 1283 }
    1292 ^2-1 = 1669263 = {3 x 431 x 1291 }
    1298 ^2-1 = 1684803 = {3 x 433 x 1297 }
    1300 ^2-1 = 1689999 = {3 x 433 x 1301 }
    1318 ^2-1 = 1737123 = {3 x 439 x 1319 }
    1328 ^2-1 = 1763583 = {3 x 443 x 1327 }
    1362 ^2-1 = 1855043 = {29 x 47 x 1361 }
    1372 ^2-1 = 1882383 = {3 x 457 x 1373 }
    1380 ^2-1 = 1904399 = {7 x 197 x 1381 }
    1382 ^2-1 = 1909923 = {3 x 461 x 1381 }
    1398 ^2-1 = 1954403 = {11 x 127 x 1399 }
    1400 ^2-1 = 1959999 = {3 x 467 x 1399 }
    1410 ^2-1 = 1988099 = {17 x 83 x 1409 }
    1438 ^2-1 = 2067843 = {3 x 479 x 1439 }
    1440 ^2-1 = 2073599 = {11 x 131 x 1439 }
    1458 ^2-1 = 2125763 = {31 x 47 x 1459 }
    1460 ^2-1 = 2131599 = {3 x 487 x 1459 }
    1470 ^2-1 = 2160899 = {13 x 113 x 1471 }
    1472 ^2-1 = 2166783 = {3 x 491 x 1471 }
    1498 ^2-1 = 2244003 = {3 x 499 x 1499 }
    1500 ^2-1 = 2249999 = {19 x 79 x 1499 }
    1510 ^2-1 = 2280099 = {3 x 503 x 1511 }
    1512 ^2-1 = 2286143 = {17 x 89 x 1511 }
    1530 ^2-1 = 2340899 = {11 x 139 x 1531 }
    1542 ^2-1 = 2377763 = {23 x 67 x 1543 }
    1554 ^2-1 = 2414915 = {5 x 311 x 1553 }
    1560 ^2-1 = 2433599 = {7 x 223 x 1559 }
    1566 ^2-1 = 2452355 = {5 x 313 x 1567 }
    1568 ^2-1 = 2458623 = {3 x 523 x 1567 }
    1570 ^2-1 = 2464899 = {3 x 523 x 1571 }
    1578 ^2-1 = 2490083 = {19 x 83 x 1579 }
    1584 ^2-1 = 2509055 = {5 x 317 x 1583 }
    1602 ^2-1 = 2566403 = {7 x 229 x 1601 }
    1622 ^2-1 = 2630883 = {3 x 541 x 1621 }
    1638 ^2-1 = 2683043 = {11 x 149 x 1637 }
    1656 ^2-1 = 2742335 = {5 x 331 x 1657 }
    1662 ^2-1 = 2762243 = {11 x 151 x 1663 }
    1670 ^2-1 = 2788899 = {3 x 557 x 1669 }
    1692 ^2-1 = 2862863 = {19 x 89 x 1693 }
    1708 ^2-1 = 2917263 = {3 x 569 x 1709 }
    1710 ^2-1 = 2924099 = {29 x 59 x 1709 }
    1732 ^2-1 = 2999823 = {3 x 577 x 1733 }
    1734 ^2-1 = 3006755 = {5 x 347 x 1733 }
    1740 ^2-1 = 3027599 = {37 x 47 x 1741 }
    1746 ^2-1 = 3048515 = {5 x 349 x 1747 }
    1752 ^2-1 = 3069503 = {17 x 103 x 1753 }
    1758 ^2-1 = 3090563 = {7 x 251 x 1759 }
    1760 ^2-1 = 3097599 = {3 x 587 x 1759 }
    1778 ^2-1 = 3161283 = {3 x 593 x 1777 }
    1782 ^2-1 = 3175523 = {13 x 137 x 1783 }
    1800 ^2-1 = 3239999 = {7 x 257 x 1801 }
    1802 ^2-1 = 3247203 = {3 x 601 x 1801 }
    1822 ^2-1 = 3319683 = {3 x 607 x 1823 }
    1830 ^2-1 = 3348899 = {31 x 59 x 1831 }
    1866 ^2-1 = 3481955 = {5 x 373 x 1867 }
    1890 ^2-1 = 3572099 = {31 x 61 x 1889 }
    1902 ^2-1 = 3617603 = {11 x 173 x 1901 }
    1908 ^2-1 = 3640463 = {23 x 83 x 1907 }
    1914 ^2-1 = 3663395 = {5 x 383 x 1913 }
    1930 ^2-1 = 3724899 = {3 x 643 x 1931 }
    1978 ^2-1 = 3912483 = {3 x 659 x 1979 }
    1980 ^2-1 = 3920399 = {7 x 283 x 1979 }
    1986 ^2-1 = 3944195 = {5 x 397 x 1987 }
    1992 ^2-1 = 3968063 = {11 x 181 x 1993 }
    2004 ^2-1 = 4016015 = {5 x 401 x 2003 }
    2018 ^2-1 = 4072323 = {3 x 673 x 2017 }
    2030 ^2-1 = 4120899 = {3 x 677 x 2029 }
    2040 ^2-1 = 4161599 = {13 x 157 x 2039 }
    2052 ^2-1 = 4210703 = {7 x 293 x 2053 }
    2070 ^2-1 = 4284899 = {19 x 109 x 2069 }
    2100 ^2-1 = 4409999 = {11 x 191 x 2099 }
    2128 ^2-1 = 4528383 = {3 x 709 x 2129 }
    2154 ^2-1 = 4639715 = {5 x 431 x 2153 }
    2160 ^2-1 = 4665599 = {17 x 127 x 2161 }
    2178 ^2-1 = 4743683 = {7 x 311 x 2179 }
    2180 ^2-1 = 4752399 = {3 x 727 x 2179 }
    2202 ^2-1 = 4848803 = {31 x 71 x 2203 }
    2214 ^2-1 = 4901795 = {5 x 443 x 2213 }
    2220 ^2-1 = 4928399 = {7 x 317 x 2221 }
    2244 ^2-1 = 5035535 = {5 x 449 x 2243 }
    2250 ^2-1 = 5062499 = {13 x 173 x 2251 }
    2252 ^2-1 = 5071503 = {3 x 751 x 2251 }
    2270 ^2-1 = 5152899 = {3 x 757 x 2269 }
    2272 ^2-1 = 5161983 = {3 x 757 x 2273 }
    2280 ^2-1 = 5198399 = {43 x 53 x 2281 }
    2282 ^2-1 = 5207523 = {3 x 761 x 2281 }
    2286 ^2-1 = 5225795 = {5 x 457 x 2287 }
    2292 ^2-1 = 5253263 = {29 x 79 x 2293 }
    2308 ^2-1 = 5326863 = {3 x 769 x 2309 }
    2334 ^2-1 = 5447555 = {5 x 467 x 2333 }
    2352 ^2-1 = 5531903 = {13 x 181 x 2351 }
    2358 ^2-1 = 5560163 = {7 x 337 x 2357 }
    2370 ^2-1 = 5616899 = {23 x 103 x 2371 }
    2390 ^2-1 = 5712099 = {3 x 797 x 2389 }
    2392 ^2-1 = 5721663 = {3 x 797 x 2393 }
    2394 ^2-1 = 5731235 = {5 x 479 x 2393 }
    2412 ^2-1 = 5817743 = {19 x 127 x 2411 }
    2418 ^2-1 = 5846723 = {41 x 59 x 2417 }
    2436 ^2-1 = 5934095 = {5 x 487 x 2437 }
    2442 ^2-1 = 5963363 = {7 x 349 x 2441 }
    2448 ^2-1 = 5992703 = {31 x 79 x 2447 }
    2460 ^2-1 = 6051599 = {23 x 107 x 2459 }
    2468 ^2-1 = 6091023 = {3 x 823 x 2467 }
    2472 ^2-1 = 6110783 = {7 x 353 x 2473 }
    2478 ^2-1 = 6140483 = {37 x 67 x 2477 }
    2502 ^2-1 = 6260003 = {41 x 61 x 2503 }
    2520 ^2-1 = 6350399 = {11 x 229 x 2521 }
    2532 ^2-1 = 6411023 = {17 x 149 x 2531 }
    2538 ^2-1 = 6441443 = {43 x 59 x 2539 }
    2544 ^2-1 = 6471935 = {5 x 509 x 2543 }
    2558 ^2-1 = 6543363 = {3 x 853 x 2557 }
    2578 ^2-1 = 6646083 = {3 x 859 x 2579 }
    2580 ^2-1 = 6656399 = {29 x 89 x 2579 }
    2590 ^2-1 = 6708099 = {3 x 863 x 2591 }
    2610 ^2-1 = 6812099 = {7 x 373 x 2609 }
    2616 ^2-1 = 6843455 = {5 x 523 x 2617 }
    2622 ^2-1 = 6874883 = {43 x 61 x 2621 }
    2632 ^2-1 = 6927423 = {3 x 877 x 2633 }
    2648 ^2-1 = 7011903 = {3 x 883 x 2647 }
    2660 ^2-1 = 7075599 = {3 x 887 x 2659 }
    2662 ^2-1 = 7086243 = {3 x 887 x 2663 }
    2670 ^2-1 = 7128899 = {17 x 157 x 2671 }
    2682 ^2-1 = 7193123 = {7 x 383 x 2683 }
    2700 ^2-1 = 7289999 = {37 x 73 x 2699 }
    2706 ^2-1 = 7322435 = {5 x 541 x 2707 }
    2720 ^2-1 = 7398399 = {3 x 907 x 2719 }
    2732 ^2-1 = 7463823 = {3 x 911 x 2731 }
    2742 ^2-1 = 7518563 = {13 x 211 x 2741 }
    2748 ^2-1 = 7551503 = {41 x 67 x 2749 }
    2778 ^2-1 = 7717283 = {7 x 397 x 2777 }
    2788 ^2-1 = 7772943 = {3 x 929 x 2789 }
    2832 ^2-1 = 8020223 = {19 x 149 x 2833 }
    2838 ^2-1 = 8054243 = {17 x 167 x 2837 }
    2842 ^2-1 = 8076963 = {3 x 947 x 2843 }
    2844 ^2-1 = 8088335 = {5 x 569 x 2843 }
    2856 ^2-1 = 8156735 = {5 x 571 x 2857 }
    2858 ^2-1 = 8168163 = {3 x 953 x 2857 }
    2860 ^2-1 = 8179599 = {3 x 953 x 2861 }
    2862 ^2-1 = 8191043 = {7 x 409 x 2861 }
    2880 ^2-1 = 8294399 = {43 x 67 x 2879 }
    2886 ^2-1 = 8328995 = {5 x 577 x 2887 }
    2898 ^2-1 = 8398403 = {13 x 223 x 2897 }
    2902 ^2-1 = 8421603 = {3 x 967 x 2903 }
    2910 ^2-1 = 8468099 = {41 x 71 x 2909 }
    2928 ^2-1 = 8573183 = {29 x 101 x 2927 }
    2940 ^2-1 = 8643599 = {17 x 173 x 2939 }
    2952 ^2-1 = 8714303 = {13 x 227 x 2953 }
    2958 ^2-1 = 8749763 = {11 x 269 x 2957 }
    2964 ^2-1 = 8785295 = {5 x 593 x 2963 }
    2972 ^2-1 = 8832783 = {3 x 991 x 2971 }
    3012 ^2-1 = 9072143 = {23 x 131 x 3011 }
    3018 ^2-1 = 9108323 = {7 x 431 x 3019 }
    3036 ^2-1 = 9217295 = {5 x 607 x 3037 }
    3038 ^2-1 = 9229443 = {3 x 1013 x 3037 }
    3040 ^2-1 = 9241599 = {3 x 1013 x 3041 }
    3042 ^2-1 = 9253763 = {17 x 179 x 3041 }
    3048 ^2-1 = 9290303 = {11 x 277 x 3049 }
    3062 ^2-1 = 9375843 = {3 x 1021 x 3061 }
    3066 ^2-1 = 9400355 = {5 x 613 x 3067 }
    3078 ^2-1 = 9474083 = {17 x 181 x 3079 }
    3084 ^2-1 = 9511055 = {5 x 617 x 3083 }
    3090 ^2-1 = 9548099 = {11 x 281 x 3089 }
    3108 ^2-1 = 9659663 = {13 x 239 x 3109 }
    3118 ^2-1 = 9721923 = {3 x 1039 x 3119 }
    3138 ^2-1 = 9847043 = {43 x 73 x 3137 }
    3162 ^2-1 = 9998243 = {29 x 109 x 3163 }
    3182 ^2-1 = 10125123 = {3 x 1061 x 3181 }
    3188 ^2-1 = 10163343 = {3 x 1063 x 3187 }
    3190 ^2-1 = 10176099 = {3 x 1063 x 3191 }
    3192 ^2-1 = 10188863 = {31 x 103 x 3191 }
    3204 ^2-1 = 10265615 = {5 x 641 x 3203 }
    3208 ^2-1 = 10291263 = {3 x 1069 x 3209 }
    3216 ^2-1 = 10342655 = {5 x 643 x 3217 }
    3222 ^2-1 = 10381283 = {11 x 293 x 3221 }
    3228 ^2-1 = 10419983 = {7 x 461 x 3229 }
    3260 ^2-1 = 10627599 = {3 x 1087 x 3259 }
    3270 ^2-1 = 10692899 = {7 x 467 x 3271 }
    3272 ^2-1 = 10705983 = {3 x 1091 x 3271 }
    3306 ^2-1 = 10929635 = {5 x 661 x 3307 }
    3308 ^2-1 = 10942863 = {3 x 1103 x 3307 }
    3318 ^2-1 = 11009123 = {31 x 107 x 3319 }
    3328 ^2-1 = 11075583 = {3 x 1109 x 3329 }
    3342 ^2-1 = 11168963 = {13 x 257 x 3343 }
    3348 ^2-1 = 11209103 = {17 x 197 x 3347 }
    3370 ^2-1 = 11356899 = {3 x 1123 x 3371 }
    3388 ^2-1 = 11478543 = {3 x 1129 x 3389 }
    3408 ^2-1 = 11614463 = {7 x 487 x 3407 }
    3414 ^2-1 = 11655395 = {5 x 683 x 3413 }
    3432 ^2-1 = 11778623 = {47 x 73 x 3433 }
    3456 ^2-1 = 11943935 = {5 x 691 x 3457 }
    3458 ^2-1 = 11957763 = {3 x 1153 x 3457 }
    3460 ^2-1 = 11971599 = {3 x 1153 x 3461 }
    3490 ^2-1 = 12180099 = {3 x 1163 x 3491 }
    3492 ^2-1 = 12194063 = {7 x 499 x 3491 }
    3498 ^2-1 = 12236003 = {13 x 269 x 3499 }
    3512 ^2-1 = 12334143 = {3 x 1171 x 3511 }
    3542 ^2-1 = 12545763 = {3 x 1181 x 3541 }
    3546 ^2-1 = 12574115 = {5 x 709 x 3547 }
    3560 ^2-1 = 12673599 = {3 x 1187 x 3559 }
    3570 ^2-1 = 12744899 = {43 x 83 x 3571 }
    3580 ^2-1 = 12816399 = {3 x 1193 x 3581 }
    3594 ^2-1 = 12916835 = {5 x 719 x 3593 }
    3612 ^2-1 = 13046543 = {23 x 157 x 3613 }
    3630 ^2-1 = 13176899 = {19 x 191 x 3631 }
    3636 ^2-1 = 13220495 = {5 x 727 x 3637 }
    3638 ^2-1 = 13235043 = {3 x 1213 x 3637 }
    3642 ^2-1 = 13264163 = {11 x 331 x 3643 }
    3660 ^2-1 = 13395599 = {7 x 523 x 3659 }
    3670 ^2-1 = 13468899 = {3 x 1223 x 3671 }
    3678 ^2-1 = 13527683 = {13 x 283 x 3677 }
    3692 ^2-1 = 13630863 = {3 x 1231 x 3691 }
    3696 ^2-1 = 13660415 = {5 x 739 x 3697 }
    3708 ^2-1 = 13749263 = {11 x 337 x 3709 }
    3710 ^2-1 = 13764099 = {3 x 1237 x 3709 }
    3738 ^2-1 = 13972643 = {37 x 101 x 3739 }
    3762 ^2-1 = 14152643 = {53 x 71 x 3761 }
    3778 ^2-1 = 14273283 = {3 x 1259 x 3779 }
    3780 ^2-1 = 14288399 = {19 x 199 x 3779 }
    3792 ^2-1 = 14379263 = {17 x 223 x 3793 }
    3798 ^2-1 = 14424803 = {29 x 131 x 3797 }
    3804 ^2-1 = 14470415 = {5 x 761 x 3803 }
    3832 ^2-1 = 14684223 = {3 x 1277 x 3833 }
    3846 ^2-1 = 14791715 = {5 x 769 x 3847 }
    3848 ^2-1 = 14807103 = {3 x 1283 x 3847 }
    3850 ^2-1 = 14822499 = {3 x 1283 x 3851 }
    3864 ^2-1 = 14930495 = {5 x 773 x 3863 }
    3882 ^2-1 = 15069923 = {11 x 353 x 3881 }
    3890 ^2-1 = 15132099 = {3 x 1297 x 3889 }
    3908 ^2-1 = 15272463 = {3 x 1303 x 3907 }
    3910 ^2-1 = 15288099 = {3 x 1303 x 3911 }
    3920 ^2-1 = 15366399 = {3 x 1307 x 3919 }
    3922 ^2-1 = 15382083 = {3 x 1307 x 3923 }
    3942 ^2-1 = 15539363 = {7 x 563 x 3943 }
    3948 ^2-1 = 15586703 = {11 x 359 x 3947 }
    3990 ^2-1 = 15920099 = {13 x 307 x 3989 }
    4008 ^2-1 = 16064063 = {19 x 211 x 4007 }
    4056 ^2-1 = 16451135 = {5 x 811 x 4057 }
    4098 ^2-1 = 16793603 = {17 x 241 x 4099 }
    4100 ^2-1 = 16809999 = {3 x 1367 x 4099 }
    4110 ^2-1 = 16892099 = {7 x 587 x 4111 }
    4134 ^2-1 = 17089955 = {5 x 827 x 4133 }
    4140 ^2-1 = 17139599 = {41 x 101 x 4139 }
    4152 ^2-1 = 17239103 = {7 x 593 x 4153 }
    4212 ^2-1 = 17740943 = {11 x 383 x 4211 }
    4228 ^2-1 = 17875983 = {3 x 1409 x 4229 }
    4270 ^2-1 = 18232899 = {3 x 1423 x 4271 }
    4282 ^2-1 = 18335523 = {3 x 1427 x 4283 }
    4284 ^2-1 = 18352655 = {5 x 857 x 4283 }
    4288 ^2-1 = 18386943 = {3 x 1429 x 4289 }
    4290 ^2-1 = 18404099 = {7 x 613 x 4289 }
    4296 ^2-1 = 18455615 = {5 x 859 x 4297 }
    4298 ^2-1 = 18472803 = {3 x 1433 x 4297 }
    4340 ^2-1 = 18835599 = {3 x 1447 x 4339 }
    4350 ^2-1 = 18922499 = {19 x 229 x 4349 }
    4358 ^2-1 = 18992163 = {3 x 1453 x 4357 }
    4392 ^2-1 = 19289663 = {23 x 191 x 4391 }
    4398 ^2-1 = 19342403 = {53 x 83 x 4397 }
    4410 ^2-1 = 19448099 = {11 x 401 x 4409 }
    4440 ^2-1 = 19713599 = {23 x 193 x 4441 }
    4442 ^2-1 = 19731363 = {3 x 1481 x 4441 }
    4448 ^2-1 = 19784703 = {3 x 1483 x 4447 }
    4450 ^2-1 = 19802499 = {3 x 1483 x 4451 }
    4452 ^2-1 = 19820303 = {61 x 73 x 4451 }
    4462 ^2-1 = 19909443 = {3 x 1487 x 4463 }
    4480 ^2-1 = 20070399 = {3 x 1493 x 4481 }
    4512 ^2-1 = 20358143 = {13 x 347 x 4513 }
    4560 ^2-1 = 20793599 = {47 x 97 x 4561 }
    4568 ^2-1 = 20866623 = {3 x 1523 x 4567 }
    4590 ^2-1 = 21068099 = {13 x 353 x 4591 }
    4592 ^2-1 = 21086463 = {3 x 1531 x 4591 }
    4596 ^2-1 = 21123215 = {5 x 919 x 4597 }
    4602 ^2-1 = 21178403 = {43 x 107 x 4603 }
    4620 ^2-1 = 21344399 = {31 x 149 x 4621 }
    4644 ^2-1 = 21566735 = {5 x 929 x 4643 }
    4648 ^2-1 = 21603903 = {3 x 1549 x 4649 }
    4658 ^2-1 = 21696963 = {3 x 1553 x 4657 }
    4662 ^2-1 = 21734243 = {59 x 79 x 4663 }
    4678 ^2-1 = 21883683 = {3 x 1559 x 4679 }
    4680 ^2-1 = 21902399 = {31 x 151 x 4679 }
    4702 ^2-1 = 22108803 = {3 x 1567 x 4703 }
    4704 ^2-1 = 22127615 = {5 x 941 x 4703 }
    4728 ^2-1 = 22353983 = {29 x 163 x 4729 }
    4734 ^2-1 = 22410755 = {5 x 947 x 4733 }
    4750 ^2-1 = 22562499 = {3 x 1583 x 4751 }
    4758 ^2-1 = 22638563 = {67 x 71 x 4759 }
    4782 ^2-1 = 22867523 = {7 x 683 x 4783 }
    4790 ^2-1 = 22944099 = {3 x 1597 x 4789 }
    4792 ^2-1 = 22963263 = {3 x 1597 x 4793 }
    4802 ^2-1 = 23059203 = {3 x 1601 x 4801 }
    4812 ^2-1 = 23155343 = {17 x 283 x 4813 }
    4818 ^2-1 = 23213123 = {61 x 79 x 4817 }
    4830 ^2-1 = 23328899 = {11 x 439 x 4831 }
    4860 ^2-1 = 23619599 = {43 x 113 x 4861 }
    4862 ^2-1 = 23639043 = {3 x 1621 x 4861 }
    4872 ^2-1 = 23736383 = {11 x 443 x 4871 }
    4890 ^2-1 = 23912099 = {67 x 73 x 4889 }
    4908 ^2-1 = 24088463 = {7 x 701 x 4909 }
    4910 ^2-1 = 24108099 = {3 x 1637 x 4909 }
    4938 ^2-1 = 24383843 = {11 x 449 x 4937 }
    4956 ^2-1 = 24561935 = {5 x 991 x 4957 }
    4970 ^2-1 = 24700899 = {3 x 1657 x 4969 }
    4972 ^2-1 = 24720783 = {3 x 1657 x 4973 }
    4986 ^2-1 = 24860195 = {5 x 997 x 4987 }
    4988 ^2-1 = 24880143 = {3 x 1663 x 4987 }
    4998 ^2-1 = 24980003 = {19 x 263 x 4999 }
    5000 ^2-1 = 24999999 = {3 x 1667 x 4999 }

    It is observed that the required condition is achieved only when ‘n’ is even.

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