On Sunday, June 19, 2022 at 5:59:30 PM UTC-7, Edward Murphy wrote:
> On 6/19/2022 5:42 PM, henh...@gmail.com wrote:
>
> > Some of the triples seem to contain "Twins"
> > 5 12 13 <------------ (12, 13)
> > 45 1012 1013 <------- the first 1000+
> > 95 4512 4513 ---------------
>
> a = m^2 - n^2, b = 2mn, c = m^2 + n^2
>
> m = 3, n = 2 -> a = 5, b = 12, c = 13
> m = 23, n = 2 -> a = 45, b = 1012, c = 1013
> m = 48, n = 47 -> a = 95, b = 4512, c = 4513
>
> and in general
> m = 5x+3, n = 5x+2 -> a = 10x + 5
> b = 50x^2 + 50x + 12
> c = 50x^2 + 50x + 13
> and x^2 + x is always even (sum of either two even numbers or two odd
> numbers), so this amounts to
> b = 100*y + 12
> c = 100*y + 13
> for some integer y, i.e. b ends in 12 and c ends in 13.
>
> A more involved followup exercise would be to identify similar patterns
> where
> b = p*10^q + r
> c = p*10^q + s
> where
> p is an integer
> q is an integer >= 2
> r and s are integers < 10^q
> and then see if multiple such patterns form any larger-scale patterns.

there's always a Twin (2-twin or 1-twin) at the end.

wow... every square (A^2) can be expressed as (C^2 - B^2)

what else do you notice ?

(load "pyth.lsp")

( 3, 4, 5 )

( 5, 12, 13 )

( 6, 8, 10 )

( 7, 24, 25 )

( 8, 15, 17 )

( 9, 12, 15 )
( 9, 40, 41 )

( 10, 24, 26 )

( 11, 60, 61 )

( 12, 16, 20 )
( 12, 35, 37 )

( 13, 84, 85 )

( 14, 48, 50 )

( 15, 20, 25 )
( 15, 36, 39 )
( 15, 112, 113 )

( 16, 30, 34 )
( 16, 63, 65 )

( 17, 144, 145 )

( 18, 24, 30 )
( 18, 80, 82 )

( 19, 180, 181 )

( 20, 21, 29 )
( 20, 48, 52 )
( 20, 99, 101 )

( 21, 28, 35 )
( 21, 72, 75 )
( 21, 220, 221 )

( 22, 120, 122 )

( 23, 264, 265 )

( 24, 32, 40 )
( 24, 45, 51 )
( 24, 70, 74 )
( 24, 143, 145 )

( 25, 60, 65 )
( 25, 312, 313 )

( 26, 168, 170 )

( 27, 36, 45 )
( 27, 120, 123 )
( 27, 364, 365 )

( 28, 45, 53 )
( 28, 96, 100 )
( 28, 195, 197 )

( 29, 420, 421 )

( 30, 40, 50 )
( 30, 72, 78 )
( 30, 224, 226 )

( 31, 480, 481 )

( 32, 60, 68 )
( 32, 126, 130 )
( 32, 255, 257 )

( 33, 44, 55 )
( 33, 56, 65 )
( 33, 180, 183 )
( 33, 544, 545 )

( 34, 288, 290 )

( 35, 84, 91 )
( 35, 120, 125 )
( 35, 612, 613 )

( 36, 48, 60 )
( 36, 77, 85 )
( 36, 105, 111 )
( 36, 160, 164 )
( 36, 323, 325 )

( 37, 684, 685 )

( 38, 360, 362 )

( 39, 52, 65 )
( 39, 80, 89 )
( 39, 252, 255 )
( 39, 760, 761 )

( 40, 42, 58 )
( 40, 75, 85 )
( 40, 96, 104 )
( 40, 198, 202 )
( 40, 399, 401 )

( 41, 840, 841 )

( 42, 56, 70 )
( 42, 144, 150 )
( 42, 440, 442 )

( 43, 924, 925 )

( 44, 117, 125 )
( 44, 240, 244 )
( 44, 483, 485 )

( 45, 60, 75 )
( 45, 108, 117 )
( 45, 200, 205 )
( 45, 336, 339 )
( 45, 1012, 1013 )

( 46, 528, 530 )

( 47, 1104, 1105 )

( 48, 55, 73 )
( 48, 64, 80 )
( 48, 90, 102 )
( 48, 140, 148 )
( 48, 189, 195 )
( 48, 286, 290 )
( 48, 575, 577 )

( 49, 168, 175 )
( 49, 1200, 1201 )

( 50, 120, 130 )
( 50, 624, 626 )

( 51, 68, 85 )
( 51, 140, 149 )
( 51, 432, 435 )
( 51, 1300, 1301 )

( 52, 165, 173 )
( 52, 336, 340 )
( 52, 675, 677 )

( 53, 1404, 1405 )

( 54, 72, 90 )
( 54, 240, 246 )
( 54, 728, 730 )

( 55, 132, 143 )
( 55, 300, 305 )
( 55, 1512, 1513 )

( 56, 90, 106 )
( 56, 105, 119 )
( 56, 192, 200 )
( 56, 390, 394 )
( 56, 783, 785 )

( 57, 76, 95 )
( 57, 176, 185 )
( 57, 540, 543 )
( 57, 1624, 1625 )

( 58, 840, 842 )

( 59, 1740, 1741 )

( 60, 63, 87 )
( 60, 80, 100 )
( 60, 91, 109 )
( 60, 144, 156 )
( 60, 175, 185 )
( 60, 221, 229 )
( 60, 297, 303 )
( 60, 448, 452 )
( 60, 899, 901 )

( 61, 1860, 1861 )

( 62, 960, 962 )

( 63, 84, 105 )
( 63, 216, 225 )
( 63, 280, 287 )
( 63, 660, 663 )
( 63, 1984, 1985 )

( 64, 120, 136 )
( 64, 252, 260 )
( 64, 510, 514 )
( 64, 1023, 1025 )

( 65, 72, 97 )
( 65, 156, 169 )
( 65, 420, 425 )
( 65, 2112, 2113 )

( 66, 88, 110 )
( 66, 112, 130 )
( 66, 360, 366 )
( 66, 1088, 1090 )

( 67, 2244, 2245 )

( 68, 285, 293 )
( 68, 576, 580 )
( 68, 1155, 1157 )

( 69, 92, 115 )
( 69, 260, 269 )
( 69, 792, 795 )
( 69, 2380, 2381 )

( 70, 168, 182 )
( 70, 240, 250 )
( 70, 1224, 1226 )

( 71, 2520, 2521 )

( 72, 96, 120 )
( 72, 135, 153 )
( 72, 154, 170 )
( 72, 210, 222 )
( 72, 320, 328 )
( 72, 429, 435 )
( 72, 646, 650 )
( 72, 1295, 1297 )

( 73, 2664, 2665 )

( 74, 1368, 1370 )

( 75, 100, 125 )
( 75, 180, 195 )
( 75, 308, 317 )
( 75, 560, 565 )
( 75, 936, 939 )
( 75, 2812, 2813 )

( 76, 357, 365 )
( 76, 720, 724 )
( 76, 1443, 1445 )

( 77, 264, 275 )
( 77, 420, 427 )
( 77, 2964, 2965 )

( 78, 104, 130 )
( 78, 160, 178 )
( 78, 504, 510 )
( 78, 1520, 1522 )

( 79, 3120, 3121 )

( 80, 84, 116 )
( 80, 150, 170 )
( 80, 192, 208 )
( 80, 315, 325 )
( 80, 396, 404 )
( 80, 798, 802 )
( 80, 1599, 1601 )

( 81, 108, 135 )
( 81, 360, 369 )
( 81, 1092, 1095 )
( 81, 3280, 3281 )

( 82, 1680, 1682 )

( 83, 3444, 3445 )

( 84, 112, 140 )
( 84, 135, 159 )
( 84, 187, 205 )
( 84, 245, 259 )
( 84, 288, 300 )
( 84, 437, 445 )
( 84, 585, 591 )
( 84, 880, 884 )
( 84, 1763, 1765 )

( 85, 132, 157 )
( 85, 204, 221 )
( 85, 720, 725 )
( 85, 3612, 3613 )

( 86, 1848, 1850 )

( 87, 116, 145 )
( 87, 416, 425 )
( 87, 1260, 1263 )
( 87, 3784, 3785 )

( 88, 105, 137 )
( 88, 165, 187 )
( 88, 234, 250 )
( 88, 480, 488 )
( 88, 966, 970 )
( 88, 1935, 1937 )

( 89, 3960, 3961 )

( 90, 120, 150 )
( 90, 216, 234 )
( 90, 400, 410 )
( 90, 672, 678 )
( 90, 2024, 2026 )

( 91, 312, 325 )
( 91, 588, 595 )
( 91, 4140, 4141 )

( 92, 525, 533 )
( 92, 1056, 1060 )
( 92, 2115, 2117 )

( 93, 124, 155 )
( 93, 476, 485 )
( 93, 1440, 1443 )
( 93, 4324, 4325 )

( 94, 2208, 2210 )

( 95, 168, 193 )
( 95, 228, 247 )
( 95, 900, 905 )
( 95, 4512, 4513 )

( 96, 110, 146 )
( 96, 128, 160 )
( 96, 180, 204 )
( 96, 247, 265 )
( 96, 280, 296 )
( 96, 378, 390 )
( 96, 572, 580 )
( 96, 765, 771 )
( 96, 1150, 1154 )
( 96, 2303, 2305 )

( 97, 4704, 4705 )

( 98, 336, 350 )
( 98, 2400, 2402 )

( 99, 132, 165 )
( 99, 168, 195 )
( 99, 440, 451 )
( 99, 540, 549 )
( 99, 1632, 1635 )
( 99, 4900, 4901 )

( 100, 105, 145 )
( 100, 240, 260 )
( 100, 495, 505 )
( 100, 621, 629 )
( 100, 1248, 1252 )
( 100, 2499, 2501 )

( 101, 5100, 5101 )

( 102, 136, 170 )
( 102, 280, 298 )
( 102, 864, 870 )
( 102, 2600, 2602 )

( 103, 5304, 5305 )

( 104, 153, 185 )
( 104, 195, 221 )
( 104, 330, 346 )
( 104, 672, 680 )
( 104, 1350, 1354 )
( 104, 2703, 2705 )

( 105, 140, 175 )
( 105, 208, 233 )
( 105, 252, 273 )
( 105, 360, 375 )
( 105, 608, 617 )
( 105, 784, 791 )
( 105, 1100, 1105 )
( 105, 1836, 1839 )
( 105, 5512, 5513 )

( 999, 1332, 1665 )
( 999, 1932, 2175 )
( 999, 4440, 4551 )
( 999, 6120, 6201 )
( 999, 13468, 13505 )
( 999, 18468, 18495 )
( 999, 55440, 55449 )
( 999, 166332, 166335 )
( 999, 499000, 499001 )

( 1000, 1050, 1450 )
( 1000, 1875, 2125 )
( 1000, 2400, 2600 )
( 1000, 3045, 3205 )
( 1000, 4950, 5050 )
( 1000, 6210, 6290 )
( 1000, 9975, 10025 )
( 1000, 12480, 12520 )
( 1000, 15609, 15641 )
( 1000, 24990, 25010 )
( 1000, 31242, 31258 )
( 1000, 49995, 50005 )
( 1000, 62496, 62504 )
( 1000, 124998, 125002 )
( 1000, 249999, 250001 )
( 1001, 2880, 3049 )
( 1001, 3432, 3575 )
( 1001, 4080, 4201 )
( 1001, 5460, 5551 )
( 1001, 6468, 6545 )
( 1001, 10200, 10249 )
( 1001, 38532, 38545 )
( 1001, 45540, 45551 )
( 1001, 71568, 71575 )
( 1001, 501000, 501001 )
( 1002, 1336, 1670 )
( 1002, 27880, 27898 )
( 1002, 83664, 83670 )
( 1002, 251000, 251002 )
( 1003, 1596, 1885 )
( 1003, 8496, 8555 )
( 1003, 29580, 29597 )
( 1003, 503004, 503005 )
( 1004, 62997, 63005 )
( 1004, 126000, 126004 )
( 1004, 252003, 252005 )
( 1005, 1340, 1675 )
( 1005, 2132, 2357 )
( 1005, 2412, 2613 )
( 1005, 6696, 6771 )
( 1005, 7504, 7571 )
( 1005, 11200, 11245 )
( 1005, 20188, 20213 )
( 1005, 33660, 33675 )
( 1005, 56108, 56117 )
( 1005, 101000, 101005 )
( 1005, 168336, 168339 )
( 1005, 505012, 505013 )

( 9999, 13332, 16665 )
( 9999, 13668, 16935 )
( 9999, 16968, 19695 )
( 9999, 44440, 45551 )
( 9999, 45360, 46449 )
( 9999, 54540, 55449 )
( 9999, 55660, 56551 )
( 9999, 137532, 137895 )
( 9999, 164832, 165135 )
( 9999, 168168, 168465 )
( 9999, 413080, 413201 )
( 9999, 494900, 495001 )
( 9999, 504900, 504999 )
( 9999, 617120, 617201 )
( 9999, 1514832, 1514865 )
( 9999, 1851468, 1851495 )
( 9999, 4544540, 4544551 )
( 9999, 5554440, 5554449 )
( 9999, 16663332, 16663335 )
( 9999, 49990000, 49990001 )

( 10000, 10500, 14500 )
( 10000, 14025, 17225 )
( 10000, 18750, 21250 )
( 10000, 24000, 26000 )
( 10000, 30450, 32050 )
( 10000, 39375, 40625 )
( 10000, 49500, 50500 )
( 10000, 62100, 62900 )
( 10000, 77805, 78445 )
( 10000, 99750, 100250 )
( 10000, 124800, 125200 )
( 10000, 156090, 156410 )
( 10000, 199875, 200125 )
( 10000, 249900, 250100 )
( 10000, 312420, 312580 )
( 10000, 390561, 390689 )
( 10000, 499950, 500050 )
( 10000, 624960, 625040 )
( 10000, 781218, 781282 )
( 10000, 999975, 1000025 )
( 10000, 1249980, 1250020 )
( 10000, 1562484, 1562516 )
( 10000, 2499990, 2500010 )
( 10000, 3124992, 3125008 )
( 10000, 4999995, 5000005 )
( 10000, 6249996, 6250004 )
( 10000, 12499998, 12500002 )
( 10000, 24999999, 25000001 )

( 10001, 364968, 365105 )
( 10001, 685032, 685105 )
( 10001, 50010000, 50010001 )

( 10002, 13336, 16670 )
( 10002, 2778880, 2778898 )
( 10002, 8336664, 8336670 )
( 10002, 25010000, 25010002 )

( 10003, 34296, 35725 )
( 10003, 1020996, 1021045 )
( 10003, 7147140, 7147147 ) --------------------
( 10003, 50030004, 50030005 )

( 10004, 13203, 16565 )
( 10004, 102297, 102785 )
( 10004, 152397, 152725 )
( 10004, 204960, 205204 )
( 10004, 305040, 305204 )
( 10004, 410103, 410225 )
( 10004, 610203, 610285 )
( 10004, 6254997, 6255005 )

( 10004, 12510000, 12510004 ) ------------------
( 10004, 25020003, 25020005 )

( 10005, 13340, 16675 )
( 10005, 17600, 20245 )
( 10005, 18576, 21099 )
( 10005, 21924, 24099 )
( 10005, 24012, 26013 )
( 10005, 28152, 29877 )
( 10005, 30744, 32331 )
( 10005, 37700, 39005 )
( 10005, 47840, 48875 )
( 10005, 59092, 59933 )
( 10005, 68672, 69397 )
( 10005, 74704, 75371 )
( 10005, 86756, 87331 )
( 10005, 94348, 94877 )
( 10005, 114840, 115275 )
( 10005, 144900, 145245 )
( 10005, 191632, 191893 )
( 10005, 222332, 222557 )
( 10005, 241684, 241891 )
( 10005, 345100, 345245 )
( 10005, 435160, 435275 )
( 10005, 575244, 575331 )
( 10005, 667296, 667371 )
( 10005, 725328, 725397 )
( 10005, 1112200, 1112245 )
( 10005, 1725848, 1725877 )
( 10005, 2001988, 2002013 )
( 10005, 2176076, 2176099 )
( 10005, 3336660, 3336675 )
( 10005, 5561108, 5561117 )
( 10005, 10010000, 10010005 )
( 10005, 16683336, 16683339 )
( 10005, 50050012, 50050013 )