• #### (3,4,5) (5,12,13) ... i can ask HAL to give me more Pythagorean triples

From henhanna@gmail.com@21:1/5 to All on Sat Jun 18 11:56:12 2022
(3,4,5) (5,12,13) ...

i can ask HAL to give me 10 more or 100 more
Pythagorean triples, but ... writing a program to search for them
is rather hard to do... (i didn't even attempt it)

https://math.stackexchange.com/questions/1386029/are-there-infinitely-many-pythagorean-triples

For all X, Y, and Z in the range 0-1000 and none equal to the
other two, I found no solutions.

Fibonacci numbers in Pythagorean triples -------
(3,4,5), (5,12,13), (16,30,34), (39,80,89), ... The middle side of each of these triangles is the sum of the three sides of the preceding triangle.

--- SoupGate-Win32 v1.05
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• From Edward Murphy@21:1/5 to henh...@gmail.com on Sun Jun 19 16:08:50 2022
On 6/18/2022 11:56 AM, henh...@gmail.com wrote:

(3,4,5) (5,12,13) ...

i can ask HAL to give me 10 more or 100 more
Pythagorean triples, but ... writing a program to search for them
is rather hard to do... (i didn't even attempt it)

Rather than searching, you could just use the method described here:
https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple
and then determine individual upper bounds on m, n, and k (above which
the hypotenuse is necessarily larger than the upper bound of your
desired range).

--- SoupGate-Win32 v1.05
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• From Edward Murphy@21:1/5 to henh...@gmail.com on Sun Jun 19 17:59:21 2022
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.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From henhanna@gmail.com@21:1/5 to Edward Murphy on Sun Jun 19 17:42:26 2022
On Sunday, June 19, 2022 at 4:08:59 PM UTC-7, Edward Murphy wrote:
On 6/18/2022 11:56 AM, henh...@gmail.com wrote:

(3,4,5) (5,12,13) ...

i can ask HAL to give me 10 more or 100 more
Pythagorean triples, but ... writing a program to search for them
is rather hard to do... (i didn't even attempt it)
Rather than searching, you could just use the method described here: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple
and then determine individual upper bounds on m, n, and k (above which
the hypotenuse is necessarily larger than the upper bound of your
desired range).

thanks.... i realized that searching is pretty easy to do...

Some of the triples seem to contain "Twins"

(define (run)
(do ((x 3 (+ 1 x))) ((> x 100))
(do ((y x (+ 1 y))) ((> y 1000000))
(let ((z (sqrt (+ (* x x) (* y y)))))
(if (integer? z)
(begin
(write x) (display " ") (write y) (display " ") (write z) (newline)))))))
(run)

3 4 5
5 12 13 <------------ (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
.........................
41 840 841
.........................

45 1012 1013 <------- the first 1000+
46 528 530
47 1104 1105 <------- the second 1000+
.........................

49 1200 1201

51 1300 1301

53 1404 1405

55 1512 1513

57 1624 1625

59 1740 1741

60 899 901
61 1860 1861

63 1984 1985

64 1023 1025

65 2112 2113

68 1155 1157

69 2380 2381

71 2520 2521

72 1295 1297

73 2664 2665

75 2812 2813

79 3120 3121

80 1599 1601

81 3280 3281

83 3444 3445

87 3784 3785

89 3960 3961

91 4140 4141

95 4512 4513 ---------------

99 4900 4901 ------------
100 105 145
100 240 260
100 495 505
100 621 629
100 1248 1252
100 2499 2501 ----------

(103, 5304, 5305)

(105, 5512, 5513)

(1003, 503004, 503005)

(1005, 505012, 505013)

#t
gosh>

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From henhanna@gmail.com@21:1/5 to henh...@gmail.com on Mon Jun 20 10:13:55 2022
On Monday, June 20, 2022 at 10:07:42 AM UTC-7, henh...@gmail.com wrote:
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)

except where A === 1, 2, 4

what else do you notice ?

( 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 )

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From henhanna@gmail.com@21:1/5 to Edward Murphy on Mon Jun 20 10:07:39 2022
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 ?

( 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 )

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Jonathan Dushoff@21:1/5 to henh...@gmail.com on Tue Jun 21 09:20:22 2022
If you're interested in these patterns, you might like this picture of the triples that I made a few years ago http://dushoff.github.io/notebook/pythagoras.html.

On Monday, June 20, 2022 at 1:13:58 PM UTC-4, henh...@gmail.com wrote:
On Monday, June 20, 2022 at 10:07:42 AM UTC-7, henh...@gmail.com wrote:
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)
except where A === 1, 2, 4
what else do you notice ?

( 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 )

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From henhanna@gmail.com@21:1/5 to Jonathan Dushoff on Tue Jun 21 10:23:12 2022
On Tuesday, June 21, 2022 at 9:20:25 AM UTC-7, Jonathan Dushoff wrote:
If you're interested in these patterns, you might like this picture of the triples that I made a few years ago http://dushoff.github.io/notebook/pythagoras.html.

thanks!
(for pedagogical reasons) or for pedagogical purposes,
students would want to see a graph showing the first 5 or the first 10 plot- points.

( 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 )

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From leflynn@21:1/5 to henh...@gmail.com on Wed Jun 22 16:53:40 2022
On Sunday, June 19, 2022 at 8:42:28 PM UTC-4, henh...@gmail.com wrote:
On Sunday, June 19, 2022 at 4:08:59 PM UTC-7, Edward Murphy wrote:
On 6/18/2022 11:56 AM, henh...@gmail.com wrote:

(3,4,5) (5,12,13) ...

i can ask HAL to give me 10 more or 100 more
Pythagorean triples, but ... writing a program to search for them
is rather hard to do... (i didn't even attempt it)
Rather than searching, you could just use the method described here: https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple
and then determine individual upper bounds on m, n, and k (above which
the hypotenuse is necessarily larger than the upper bound of your
desired range).
thanks.... i realized that searching is pretty easy to do...

Some of the triples seem to contain "Twins"

You do realize that one class of triples can be constructed by taking any odd number a=n
together with b=(n*n-1)/2 and c=b+1 .
L. Flynn

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Jonathan Dushoff@21:1/5 to henh...@gmail.com on Fri Jun 24 11:14:50 2022
Thanks!

Doing that made me notice that I put the code together in a big hurry, and was only finding triples where c is itself a sum of squares (which excludes things like (9, 12, 15) where c is the product of a sum of square and and a non-SOS factor). The whole
thing has become a bit of a rabbit hole for me, but here is a possibly pedagogical plot.

http://dushoff.github.io/notebook/outputs/rp.newpyth.Rout.pdf

On Tuesday, June 21, 2022 at 1:23:15 PM UTC-4, henh...@gmail.com wrote:
On Tuesday, June 21, 2022 at 9:20:25 AM UTC-7, Jonathan Dushoff wrote:
If you're interested in these patterns, you might like this picture of the triples that I made a few years ago http://dushoff.github.io/notebook/pythagoras.html.
thanks!
(for pedagogical reasons) or for pedagogical purposes,
students would want to see a graph showing the first 5 or the first 10 plot- points.

( 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 )
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( 96, 110, 146 )
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( 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 )
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( 100, 621, 629 )
( 100, 1248, 1252 )
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( 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 )
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( 105, 140, 175 )
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( 1000, 15609, 15641 )
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--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From henhanna@gmail.com@21:1/5 to Jonathan Dushoff on Fri Jun 24 19:09:20 2022
All possible pythagorean triples, visualized

3,086,907 views -- May 26, 2017

i found this 20 min. ago....

watching these clips by 3Blue1Brown (4.59M subscribers) ...
i often wonder ... Will these clips make Martin-Gardner (like) articles obsolete (forgotten) ?

On Friday, June 24, 2022 at 11:14:52 AM UTC-7, Jonathan Dushoff wrote:
Thanks!

Doing that made me notice that I put the code together in a big hurry, and was only finding triples where c is itself a sum of squares (which excludes things like (9, 12, 15) where c is the product of a sum of square and and a non-SOS factor). The
whole thing has become a bit of a rabbit hole for me, but here is a possibly pedagogical plot.

http://dushoff.github.io/notebook/outputs/rp.newpyth.Rout.pdf
On Tuesday, June 21, 2022 at 1:23:15 PM UTC-4, henh...@gmail.com wrote:
On Tuesday, June 21, 2022 at 9:20:25 AM UTC-7, Jonathan Dushoff wrote:
If you're interested in these patterns, you might like this picture of the triples that I made a few years ago http://dushoff.github.io/notebook/pythagoras.html.
thanks!
(for pedagogical reasons) or for pedagogical purposes,
students would want to see a graph showing the first 5 or the first 10 plot- points.

( 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 )
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( 1000, 124998, 125002 )
( 1000, 249999, 250001 )
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( 1001, 3432, 3575 )
( 1001, 4080, 4201 )
( 1001, 5460, 5551 )
( 1001, 6468, 6545 )
( 1001, 10200, 10249 )
( 1001, 38532, 38545 )
( 1001, 45540, 45551 )
( 1001, 71568, 71575 )
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( 1002, 27880, 27898 )
( 1002, 83664, 83670 )
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( 1003, 8496, 8555 )
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( 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 )

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)