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= re: talk:Tree of primitive Pythagorean triples = from talk:Tree of primitive Pythagorean triples&direction=prev&oldid=906435953 Gangleri 217.86.249.144 (talk) 12:15, 16 July 2019 (UTC)

Dual representation for the tree
Hi! The table above combines the representation of the primitive Pythagorean triplets with the values of the primitive pair of integers generating them.

The table includes as additional root the trivial triplet (1, 0, 1) which is generated by the (m, n) pair (1, 0). Please note that all (m, n) pairs are written with m < n which is the opposite of the requirement described in the article.

The alternative calculations of the childs of any Pythagorean triplet are as follow:
 * up: (mnew, nnew) = (2m-n, m)
 * Pell pair: (mnew, nnew) = (2m+n, m)
 * for (m, n) other then (1, 0) down: (mnew, nnew) = (m+2n, n)
 * note: from (1, 0) both the up path and the "Pell" path generate the same main root value (2, 1)

The calculations of the triplet childs might be simpler as using matrix calculations. Please note that both the up way and the "Pell" way are preserving m, the greater value of the (m, n) pair, while the down way preserves n, the smaller value of the (m, n) pair.

Additional remark on the Pell numbers: The sequence consists of values of the hypotenuse of a Pythagorean triplet followed by its circumference (a+b+c). It starts with 1 and 2. This pair generates the triplet (3, 4, 5) so both 5 and 12 = 3 + 4 + 5 can be added to the sequence. By using (2, 5) one generates the triplet (21, 20, 29) so both 29 and 70 = 21 + 20 +29 can be added to the sequence. Nevertheless (5, 12) is the next pair to be used on the "Pell" path.

Best regards Gangleri
 * The triplet (0, 1, 1) generated by the (1, 0) pair does not have a parent Pythagorean triplet.
 * In order to calculate the parent of a distinct arbitrary primitive Pythagorean triplet (a, b, c) first one must identify the generating (m, n) pair which is generating the examined Pythagorean triplet. The pair is identified via
 * m=square root(c²-a²)/2 and because m is not zero for triples other then (0, 1, 1) n=b/2m
 * algorithm:
 * note: the case for n=0 is discussed above
 * if m-2n=0 then the analysed Pythagorean triplet is the main root Pythagorean triplet (3, 4, 5)
 * if m-2n<0 then (mnew, nnew) = (n, 2n-m) and the way was "up"
 * else if n>m-2n then (mnew, nnew) = (n, m-2n) and the way was the "Pell" way
 * else (mnew, nnew) = (m-2n, n) and the way was "down".
 * Regards Gangleri — Preceding unsigned comment added by 2A02:810D:41C0:134:5D5D:FEFD:1D74:D8AF (talk) 12:22, 6 July 2019 (UTC)

Switched to (m, n) with m > n representation. Regards Gangleri 13:46, 7 July 2019 (UTC)

Hopping on the Pell path
Any two consecutive numbers of the Pell series Pk = n and Pk + 1 = m generate a Pythagorean triplet (a, b, c). The (m, n) pair is used to generate (m² - n², 2mn, m² + n²) with the additional property that | a - b | = 1. Take p = 2m² + 2mn the circumference of the triangle and q = m² + n² the hypotenuse of the triangle. With (p, q) a new Pythagorean triplet can be generated having an interesting circumference and an interesting hypotenuse. Now consider the iteration pnew = 2p² + 2pq the circumference and qnew = p² + q² the hypotenuse. These are two different consecutive numbers of the Pell series so they generat a Pythagorean triplet with | a - b | = 1.

Examples:
 * (P2, P1) i.e. (2, 1) iterates to
 * (P4, P3) i.e. (12, 5) iterates to
 * (P8, P7) i.e. (408 169) iterates to
 * (P16, P15) i.e. (470832 195025) iterates to
 * (P32, P31) i.e. (627013566048 259717522849)

Best regards Gangleri 18:42, 7 July 2019 (UTC) — Preceding unsigned comment added by 2A02:810D:41C0:134:F81F:7483:C15A:1780 (talk)
 * What are neither coincidences nor obvious for any two consecutive numbers of the Pell series (Pn, Pn - 1) are properties for their generated Pythagorean triangle (a, b, c)
 * i) c is a Pell number
 * ii) a + b + c is a Pell number too
 * iii) a + b + c is the successor of c in the Pell series
 * Best regards Gangleri 20:33, 7 July 2019 (UTC)

Note that the predecessor of the Pell number c can be calculated directly as a + b - c. So (c, a + b - c) can be used as a pair of two consecutive Pell numbers witch differs from the original (m, n) if that was not (1, 0).

Because of this relation an additional hopping path can be used in order to find two consecutive Pell numbers. From Pk = n and Pk + 1 = m we use the hypotenuse c of the generated Pythagorean triangle and its predecessor a + b - c using the formulas
 * mnew = m² + n²
 * nnew = 2mn - 2n²

Examples:
 * (P2, P1) i.e. (2, 1) iterates to
 * (P3, P2) i.e. (5, 2) iterates to
 * (P5, P4) i.e. (29 12) iterates to
 * (P9, P8) i.e. (985 408) iterates to
 * (P17, P16) i.e. (1136689 470832)

Conclusion: Together with the original Pell iteration Pn = Pn - 1 + 2Pn - 2 (for n >=2 ) there are tree paths from one pair of consecutive Pell numbers other then (1, 0) to another pair of consecutive Pell numbers.

Best regards Gangleri 09:37, 8 July 2019 (UTC)

Trees and their objects
Each primitive Pythagorean triplet can be identified by one of the following key values:
 * a) generating method for an arbitrary primitive Pythagorean triplet and the path to that triplet; because only two exhausting methods are known this could be either one of
 * a1) Berggrens's method and the path to the triplet in the Berggrens's tree as "0,11213" used below
 * a2) Price's method and the path to the triplet in the Price's tree as "1,2232" used below
 * or both
 * b) triplet values as (a, b, c) where a and c are odd integer numbers and b is an even integer number which might be zero; gcd(a, b, c) = 1 and a, b and c fullfill the Pythagorean equation a² + b² = c²; triplets not obeing the Pythagorean equation are not discussed in the following
 * c) the generating pair (m, n) where m and n are integer numbers m > n >= 0 and gcd(m, n) = 1 generating the primitive Pythagorean triplet (a, b, c) with the set of formulas 1 ; any artitrary pair (m, n) with the properties from above identifies a unique primitive Pythagorean triplet
 * d) the generating pair (u, v) where u and v are integer numbers u > v and both values are odd and gcd(u, v) = 1 and they generate the same primitive Pythagorean triplet (a, b, c) with the set of formulas 2 ; any artitrary pair (m, n) with the properties from above identifies a unique primitive Pythagorean triplet
 * e) the pair (p, l) formed by the numerical values of the perimeter and "lead" for a triangle formed by a primitive Pythagorean triplet (a, b, c) ; the lead of the sum of the length of the two legs of the associated Pythagorean triangle to the length of the hypotenuse of the Pythagorean triangle; please note the relations p = a + b + c and l = a + b - c as well as c = (p - l) / 2 ; both p and l are even numbers; there are pairs of (p, l) which do neiter relate to primitive Pythagorean triplet nor to general Pythagorean triplet; this means that (p, l) can not be selected randomly

The necessary calculations / formulas to determine the four other associated key values when one key is known will be described in the detail subparagraphs. Notes: The perimeters of two triangles associated to two different Pythagorean triplets might have the same values as can be seen searching for the hashtags "#p1a" to "#p3a" and "#p1b" to "#p5b" in the following two tables.

Berggrens's tree
Notes: added some (u, v) pairs ; intermittent saving regards Gangleri 16:32, 13 July 2019 (UTC)

details to Berggrens's tree
to be continued 2A02:810D:41C0:134:6C5D:2ACF:DA6:D533 (talk) 18:07, 12 July 2019 (UTC)

4 4 12 9 24 16 40 25 60 36 84 49 204 289 160 256 140 196 644 529 924 1089 240 576 104 169 572 484 780 900 168 441 88 121 396 324 900 625 1692 2209 880 1600 572 676 2132 1681 3276 3969 1056 2304 152 361 1292 1156 1596 1764 216 729 60 100 340 289 816 576 1496 1936 740 1369 460 529 1656 1296 2576 3136 860 1849 96 256 928 841 1120 1225 132 484 48 64 208 169 468 324 828 529 1764 2401 1144 1936 884 1156 3740 3025 5508 6561 1560 3600 464 841 2900 2500 3828 4356 720 2025 304 361 1140 900 2460 1681 4740 6241 2584 4624 1748 2116 6716 5329 10212 12321 3192 7056 560 1225 4340 3844 5460 6084 816 2601 84 196 700 625 1800 1296 3200 4096 1484 2809 868 961 2976 2304 4712 5776 1652 3481 120 400 1480 1369 1720 1849 156 676 28 49 168 144 408 289 748 484 1564 2116 984 1681 744 961 3100 2500 4588 5476 1320 3025 364 676 2340 2025 3068 3481 560 1600 224 256 800 625 1700 1156 3300 4356 1824 3249 1248 1521 4836 3844 7332 8836 2272 5041 420 900 3180 2809 4020 4489 616 1936 44 121 440 400 1160 841 2040 2601 924 1764 528 576 1776 1369 2832 3481 1012 2116 60 225 840 784 960 1024 76 361 20 25 80 64 176 121 308 196 476 289 1092 1521 792 1296 660 900 2940 2401 4260 5041 1144 2704 432 729 2484 2116 3348 3844 688 1849 336 441 1428 1156 3196 2209 6052 7921 3192 5776 2100 2500 7900 6241 12100 14641 3864 8464 592 1369 4884 4356 6068 6724 848 2809 180 324 1116 961 2728 1936 4960 6400 2412 4489 1476 1681 5248 4096 8200 10000 2772 5929 280 784 2856 2601 3416 3721 380 1444 120 144 456 361 988 676 1716 1089 3692 5041 2432 4096 1900 2500 8100 6561 11900 14161 3344 7744 1032 1849 6364 5476 8428 9604 1608 4489 696 841 2668 2116 5796 3969 11132 14641 6032 10816 4060 4900 15540 12321 23660 28561 7424 16384 1272 2809 9964 8836 12508 13924 1848 5929 220 484 1716 1521 4368 3136 7800 10000 3652 6889 2156 2401 7448 5776 11760 14400 4092 8649 320 1024 3776 3481 4416 4761 420 1764 36 81 288 256 736 529 1380 900 2852 3844 1760 3025 1312 1681 5412 4356 8036 9604 2336 5329 612 1156 4012 3481 5236 5929 936 2704 360 400 1240 961 2604 1764 5084 6724 2840 5041 1960 2401 7644 6084 11564 13924 3560 7921 684 1444 5092 4489 6460 7225 1008 3136 52 169 624 576 1680 1225 2928 3721 1300 2500 728 784 2408 1849 3864 4761 1404 2916 68 289 1088 1024 1224 1296 84 441 8 16 56 49 140 100 260 169 416 256 936 1296 660 1089 540 729 2376 1936 3456 4096 940 2209 336 576 1968 1681 2640 3025 532 1444 252 324 1044 841 2320 1600 4408 5776 2340 4225 1548 1849 5848 4624 8944 10816 2844 6241 448 1024 3648 3249 4544 5041 644 2116 120 225 780 676 1924 1369 3484 4489 1680 3136 1020 1156 3604 2809 5644 6889 1920 4096 184 529 1932 1764 2300 2500 248 961 72 81 252 196 532 361 912 576 1976 2704 1316 2209 1036 1369 4440 3600 6512 7744 1820 4225 576 1024 3520 3025 4672 5329 900 2500 396 484 1540 1225 3360 2304 6440 8464 3476 6241 2332 2809 8904 7056 13568 16384 4268 9409 720 1600 5680 5041 7120 7921 1044 3364 136 289 1020 900 2580 1849 4620 5929 2176 4096 1292 1444 4484 3481 7068 8649 2448 5184 200 625 2300 2116 2700 2916 264 1089 12 36 132 121 352 256 672 441 1376 1849 836 1444 616 784 2520 2025 3752 4489 1100 2500 276 529 1840 1600 2392 2704 420 1225 156 169 520 400 1080 729 2120 2809 1196 2116 832 1024 3264 2601 4928 5929 1508 3364 300 625 2200 1936 2800 3136 444 1369 16 64 240 225 660 484 1140 1444 496 961 272 289 884 676 1428 1764 528 1089 20 100 380 361 420 441 24 144


 * added sorted (b, n) pairs for the Berggrens's tree; regards Gangleri 217.86.249.144 (talk) 13:14, 16 July 2019 (UTC)

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1_0_1|0	1 0 1	 	1_0_1|0	1 0 1		1 1005_2132_2357|0,111221	1005 2132 2357		1005_2132_2357|1,31221	1005 2132 2357		1 1007_1224_1585|0,123332	1007 1224 1585		1007_1224_1585|1,11122	1007 1224 1585		1 101_5100_5101|1,31131	101 5100 5101		1015_192_1033|1,2332	1015 192 1033		10205_10212_14437|0,112222	10205 10212 14437		1023_64_1025|1,3332	1023 64 1025		1025_528_1153|0,133323	1025 528 1153		103_5304_5305|1,31133	103 5304 5305		1037_1716_2005|0,12231	1037 1716 2005		1045_4452_4573|1,22231	1045 4452 4573		105_208_233|0,1121	105 208 233		105_208_233|1,12111	105 208 233		1 105_5512_5513|1,31311	105 5512 5513		105_608_617|1,21111	105 608 617		105_88_137|0,1112	105 88 137		105_88_137|1,3211	105 88 137		1 1053_3196_3365|0,121211	1053 3196 3365		1065_2408_2633|0,123321	1065 2408 2633		107_5724_5725|1,31313	107 5724 5725		1071_1840_2129|0,133131	1071 1840 2129		1071_1840_2129|1,21323	1071 1840 2129		1 1071_7040_7121|1,32233	1071 7040 7121		1073_264_1105|0,132333	1073 264 1105		1075_612_1237|0,12313	1075 612 1237		1079_3360_3529|0,132211	1079 3360 3529		1079_3360_3529|1,13223	1079 3360 3529		1 1085_132_1093|1,33321	1085 132 1093		109_5940_5941|1,31331	109 5940 5941		11_60_61|0,11111	11 60 61		11_60_61|1,13	11 60 61		1 1105_1968_2257|0,131131	1105 1968 2257		1107_476_1205|1,12212	1107 476 1205		111_6160_6161|1,31333	111 6160 6161		111_680_689|1,2133	111 680 689		11115_12508_16733|0,122232	11115 12508 16733		1127_936_1465|0,131112	1127 936 1465		113_6384_6385|1,33111	113 6384 6385		1131_340_1181|1,22112	1131 340 1181		1147_204_1165|1,21312	1147 204 1165		115_252_277|0,1321	115 252 277		115_252_277|1,1213	115 252 277		1 115_6612_6613|1,33113	115 6612 6613		11523_11564_16325|0,123222	11523 11564 16325		1155_1292_1733|0,13232	1155 1292 1733		1155_68_1157|1,11112	1155 68 1157		1157_3876_4045|1,23231	1157 3876 4045		1159_1680_2041|1,11323	1159 1680 2041		1161_560_1289|0,11223	1161 560 1289		11661_11900_16661|0,122122	11661 11900 16661		117_44_125|0,1133	117 44 125		117_44_125|1,1121	117 44 125		1 117_6844_6845|1,33131	117 6844 6845		1173_1036_1565|0,13212	1173 1036 1565		1175_792_1417|0,121113	1175 792 1417		1175_792_1417|1,32122	1175 792 1417		1 1189_420_1261|0,133133	1189 420 1261		1189_420_1261|1,23321	1189 420 1261		1 119_120_169|0,122	119 120 169		119_120_169|1,122	119 120 169		1 119_7080_7081|1,33133	119 7080 7081		11999_11760_16801|0,122322	11999 11760 16801		1207_2376_2665|0,131121	1207 2376 2665		1207_2376_2665|1,22123	1207 2376 2665		1 121_7320_7321|1,33311	121 7320 7321		12141_12100_17141|0,121222	12141 12100 17141		1219_1140_1669|0,133312	1219 1140 1669		123_7564_7565|1,33313	123 7564 7565		123_836_845|1,21113	123 836 845		1235_1932_2293|0,131331	1235 1932 2293		1239_1520_1961|1,11322	1239 1520 1961		1241_2520_2809|0,133121	1241 2520 2809		1245_3332_3557|1,33231	1245 3332 3557		1247_504_1345|1,32132	1247 504 1345		125_7812_7813|1,33331	125 7812 7813		12525_11132_16757|0,122212	12525 11132 16757		1265_1248_1777|0,11322	1265 1248 1777		1269_740_1469|0,111313	1269 740 1469		1269_740_1469|1,22321	1269 740 1469		1 127_8064_8065|1,33333	127 8064 8065		1275_2668_2957|0,12221	1275 2668 2957		1275_988_1613|1,23212	1275 988 1613		1287_4816_4985|1,31233	1287 4816 4985		1287_6784_6905|1,22233	1287 6784 6905		129_920_929|1,2311	129 920 929		1295_72_1297|1,11132	1295 72 1297		13_84_85|0,111111	13 84 85		13_84_85|1,31	13 84 85		1 1305_592_1433|0,12123	1305 592 1433		1309_2820_3109|1,33221	1309 2820 3109		1311_1360_1889|1,21322	1311 1360 1889		1311_2200_2561|0,133231	1311 2200 2561		1323_836_1565|0,133113	1323 836 1565		1325_1092_1717|0,121112	1325 1092 1717		1325_1092_1717|1,32321	1325 1092 1717		1 133_156_205|0,1332	133 156 205		133_156_205|1,2121	133 156 205		1 1333_444_1405|0,133233	1333 444 1405		1343_2976_3265|0,112321	1343 2976 3265		1343_2976_3265|1,12223	1343 2976 3265		1 1349_2340_2701|0,113131	1349 2340 2701		1349_2340_2701|1,23221	1349 2340 2701		1 135_352_377|0,13311	135 352 377		135_352_377|1,223	135 352 377		1 1357_1476_2005|0,12132	1357 1476 2005		1357_1476_2005|1,32221	1357 1476 2005		1 13575_13568_19193|0,132222	13575 13568 19193		1363_684_1525|0,12323	1363 684 1525		1365_1892_2333|1,22221	1365 1892 2333		1387_2484_2845|0,121131	1387 2484 2845		1395_532_1493|0,131133	1395 532 1493		1403_1596_2125|0,111232	1403 1596 2125		1407_2024_2465|1,13123	1407 2024 2465		141_1100_1109|1,21131	141 1100 1109		1419_380_1469|0,121333	1419 380 1469		1419_380_1469|1,12312	1419 380 1469		1 1425_1312_1937|0,12312	1425 1312 1937		143_24_145|0,133333	143 24 145		143_24_145|1,132	143 24 145		1 1431_1040_1769|1,21222	1431 1040 1769		1443_76_1445|1,11312	1443 76 1445		1449_1720_2249|0,112332	1449 1720 2249		145_408_433|0,11311	145 408 433		145_408_433|1,2211	145 408 433		1 1455_4592_4817|1,31223	1455 4592 4817		1463_2784_3145|1,12323	1463 2784 3145		147_1196_1205|1,2313	147 1196 1205		1475_1428_2053|0,133322	1475 1428 2053		1479_880_1721|0,111213	1479 880 1721		1479_880_1721|1,11222	1479 880 1721		1 1491_2300_2741|0,132331	1491 2300 2741		1495_1848_2377|1,13122	1495 1848 2377		1495_6528_6697|1,23233	1495 6528 6697		15_112_113|1,33	15 112 113		15_8_17|0,13	15 8 17		15_8_17|1,2	15 8 17		1 1501_2940_3301|0,121121	1501 2940 3301		1519_720_1681|0,13223	1519 720 1681		1519_720_1681|1,11232	1519 720 1681		1 1525_1548_2173|0,13122	1525 1548 2173		153_104_185|0,11113	153 104 185		153_104_185|1,11211	153 104 185		1 1537_984_1825|0,113113	1537 984 1825		1539_3100_3461|0,113121	1539 3100 3461		155_468_493|0,11211	155 468 493		155_468_493|1,12113	155 468 493		1 1551_560_1649|0,113133	1551 560 1649		1551_560_1649|1,21232	1551 560 1649		1 15543_7424_17225|0,122223	15543 7424 17225		1577_3264_3625|0,133221	1577 3264 3625		159_1400_1409|1,21133	159 1400 1409		1591_240_1609|1,21332	1591 240 1609		1593_1376_2105|0,133112	1593 1376 2105		1599_80_1601|1,11332	1599 80 1601		161_240_289|0,13331	161 240 289		161_240_289|1,32111	161 240 289		1 1615_4368_4657|0,122311	1615 4368 4657		1643_924_1885|0,113313	1643 924 1885		1643_924_1885|1,22212	1643 924 1885		1 1647_1496_2225|0,111312	1647 1496 2225		1647_1496_2225|1,22122	1647 1496 2225		1 165_1508_1517|1,2331	165 1508 1517		165_52_173|0,1233	165 52 173		165_52_173|1,1321	165 52 173		1 165_532_557|0,13211	165 532 557		165_532_557|1,1231	165 532 557		1 1653_3604_3965|0,131321	1653 3604 3965		1659_2900_3341|0,112131	1659 2900 3341		1679_2400_2929|1,13323	1679 2400 2929		1695_6272_6497|1,33233	1695 6272 6497		17_144_145|1,111	17 144 145		1705_1032_1993|0,12213	1705 1032 1993		171_140_221|0,11112	171 140 221		171_140_221|1,1212	171 140 221		1 1739_420_1789|0,122333	1739 420 1789		1749_860_1949|0,111323	1749 860 1949		175_288_337|0,1231	175 288 337		175_288_337|1,323	175 288 337		1 1755_1748_2477|0,11222	1755 1748 2477		1763_84_1765|1,13112	1763 84 1765		1767_1144_2105|0,112113	1767 1144 2105		177_1736_1745|1,21311	177 1736 1745		1775_2208_2833|1,13322	1775 2208 2833		1785_688_1913|0,121133	1785 688 1913		1817_2856_3385|0,121331	1817 2856 3385		1827_1564_2405|0,113112	1827 1564 2405		183_1856_1865|1,2333	183 1856 1865		185_672_697|0,133111	185 672 697		1853_5796_6085|0,122211	1853 5796 6085		1863_3016_3545|1,23123	1863 3016 3545		1869_3740_4181|0,112121	1869 3740 4181		187_84_205|0,1123	187 84 205		187_84_205|1,2112	187 84 205		1 1885_1692_2533|0,111212	1885 1692 2533		1887_6016_6305|1,33223	1887 6016 6305		1887_616_1985|0,113233	1887 616 1985		189_340_389|0,11131	189 340 389		189_340_389|1,1221	189 340 389		1 19_180_181|1,113	19 180 181		1909_3180_3709|0,113231	1909 3180 3709		1911_440_1961|1,22132	1911 440 1961		1917_2156_2885|0,12232	1917 2156 2885		1935_88_1937|1,13132	1935 88 1937		1943_1824_2665|1,12322	1943 1824 2665		1947_1196_2285|0,133213	1947 1196 2285		195_2108_2117|1,21313	195 2108 2117		195_28_197|1,312	195 28 197		195_748_773|0,113111	195 748 773		195_748_773|1,2213	195 748 773		1 1961_720_2089|0,112133	1961 720 2089		1971_2300_3029|0,131332	1971 2300 3029		1975_2808_3433|1,31123	1975 2808 3433		1995_1012_2237|0,113323	1995 1012 2237		2001_1960_2801|0,12322	2001 1960 2801		2001_3520_4049|0,132131	2001 3520 4049		2013_1316_2405|0,132113	2013 1316 2405		2015_1632_2593|1,12222	2015 1632 2593		203_396_445|0,11121	203 396 445		203_396_445|1,3213	203 396 445		1 2035_828_2197|1,32212	2035 828 2197		2037_4484_4925|0,132321	2037 4484 4925		205_828_853|0,112111	205 828 853		205_828_853|1,12131	205 828 853		1 2059_2100_2941|0,12122	2059 2100 2941		2067_644_2165|0,131233	2067 644 2165		207_224_305|0,1132	207 224 305		207_224_305|1,322	207 224 305		1 2071_5760_6121|1,23223	2071 5760 6121		2077_1764_2725|0,112112	2077 1764 2725		2079_2600_3329|1,31122	2079 2600 3329		209_120_241|0,1313	209 120 241		209_120_241|1,13211	209 120 241		1 21_20_29|0,12	21 20 29		21_20_29|1,21	21 20 29		1 21_220_221|1,131	21 220 221		2107_276_2125|1,23112	2107 276 2125		2109_940_2309|0,131123	2109 940 2309		2115_92_2117|1,13312	2115 92 2117		213_2516_2525|1,21331	213 2516 2525		2135_1248_2473|1,13222	2135 1248 2473		2139_1900_2861|0,12212	2139 1900 2861		215_912_937|0,132111	215 912 937		215_912_937|1,1233	215 912 937		1 217_456_505|0,1221	217 456 505		217_456_505|1,21211	217 456 505		1 2175_2392_3233|0,133132	2175 2392 3233		2175_2392_3233|1,23122	2175 2392 3233		1 2183_1056_2425|0,111223	2183 1056 2425		2183_1056_2425|1,13232	2183 1056 2425		1 2201_2040_3001|0,113312	2201 2040 3001		221_60_229|0,11333	221 60 229		221_60_229|1,3121	221 60 229		1 2225_3648_4273|0,131231	2225 3648 4273		2231_4440_4969|0,132121	2231 4440 4969		2247_5504_5945|1,22223	2247 5504 5945		225_272_353|0,13332	225 272 353		2255_672_2353|1,12232	2255 672 2353		2279_480_2329|1,12332	2279 480 2329		2291_2700_3541|0,132332	2291 2700 3541		2295_3248_3977|1,31323	2295 3248 3977		23_264_265|1,133	23 264 265		2303_96_2305|1,13332	2303 96 2305		231_160_281|0,111113	231 160 281		231_160_281|1,222	231 160 281		1 231_2960_2969|1,21333	231 2960 2969		231_520_569|0,13321	231 520 569		231_520_569|1,2123	231 520 569		1 2323_4836_5365|0,113221	2323 4836 5365		2325_2332_3293|0,13222	2325 2332 3293		2325_4012_4637|0,123131	2325 4012 4637		2331_1300_2669|0,123313	2331 1300 2669		2343_1976_3065|0,132112	2343 1976 3065		23661_23660_33461|0,122222	23661 23660 33461		2379_1100_2621|0,133123	2379 1100 2621		2407_3024_3865|1,31322	2407 3024 3865		2409_2120_3209|0,133212	2409 2120 3209		2415_5248_5777|0,121321	2415 5248 5777		2415_5248_5777|1,32223	2415 5248 5777		1 2419_900_2581|0,132133	2419 900 2581		2449_2640_3601|0,131132	2449 2640 3601		245_1188_1213|1,2231	245 1188 1213		2451_700_2549|1,32312	2451 700 2549		2457_3776_4505|0,122331	2457 3776 4505		247_96_265|0,11133	247 96 265		247_96_265|1,232	247 96 265		1 249_3440_3449|1,23111	249 3440 3449		2499_100_2501|1,31112	2499 100 2501		25_312_313|1,311	25 312 313		2511_2800_3761|0,133232	2511 2800 3761		253_204_325|0,111112	253 204 325		253_204_325|1,21121	253 204 325		1 2537_816_2665|0,112233	2537 816 2665		255_1288_1313|1,12133	255 1288 1313		255_32_257|1,332	255 32 257		2575_4992_5617|1,32323	2575 4992 5617		2583_1144_2825|0,121123	2583 1144 2825		259_660_709|0,133311	259 660 709		259_660_709|1,32113	259 660 709		1 2607_2576_3665|0,111322	2607 2576 3665		261_380_461|0,133331	261 380 461		261_380_461|1,12121	261 380 461		1 2613_1484_3005|0,112313	2613 1484 3005		2619_4340_5069|0,112231	2619 4340 5069		2623_936_2785|0,123133	2623 936 2785		2639_3720_4561|1,33123	2639 3720 4561		265_1392_1417|1,22111	265 1392 1417		2665_1272_2953|0,12223	2665 1272 2953		267_3956_3965|1,23113	267 3956 3965		2675_5412_6037|0,123121	2675 5412 6037		2695_312_2713|1,23132	2695 312 2713		27_364_365|1,313	27 364 365		2703_104_2705|1,31132	2703 104 2705		2727_4736_5465|1,22323	2727 4736 5465		273_136_305|0,1323	273 136 305		273_136_305|1,31211	273 136 305		1 273_736_785|0,12311	273 736 785		2745_848_2873|0,121233	2745 848 2873		2747_1404_3085|0,123323	2747 1404 3085		275_252_373|0,1312	275 252 373		275_252_373|1,3212	275 252 373		1 2759_3480_4441|1,33122	2759 3480 4441		2769_1760_3281|0,123113	2769 1760 3281		2775_5848_6473|0,131221	2775 5848 6473		279_440_521|0,11331	279 440 521		279_440_521|1,1123	279 440 521		1 2805_3068_4157|0,113132	2805 3068 4157		285_4508_4517|1,23131	285 4508 4517		285_68_293|0,12333	285 68 293		285_68_293|1,3321	285 68 293		1 287_816_865|0,111311	287 816 865		287_816_865|1,1223	287 816 865		1 2871_4480_5321|1,23323	2871 4480 5321		2881_1320_3169|0,113123	2881 1320 3169		2891_540_2941|1,22312	2891 540 2941		29_420_421|1,331	29 420 421		2905_2832_4057|0,113322	2905 2832 4057		2911_1680_3361|0,131313	2911 1680 3361		2911_1680_3361|1,31222	2911 1680 3361		1 2915_108_2917|1,31312	2915 108 2917		2937_3416_4505|0,121332	2937 3416 4505		295_1728_1753|1,2233	295 1728 1753		2967_1456_3305|1,31232	2967 1456 3305		297_304_425|0,1122	297 304 425		2987_4884_5725|0,121231	2987 4884 5725		299_180_349|0,1213	299 180 349		299_180_349|1,12112	299 180 349		1 2993_1824_3505|0,113213	2993 1824 3505		3_4_5|0,1	3 4 5		3_4_5|1	3 4 5		1 3007_4224_5185|1,33323	3007 4224 5185		301_900_949|0,111211	301 900 949		301_900_949|1,3231	301 900 949		1 303_5096_5105|1,23133	303 5096 5105		3045_5092_5933|0,123231	3045 5092 5933		305_1848_1873|1,12311	305 1848 1873		3055_1008_3217|0,123233	3055 1008 3217		31_480_481|1,333	31 480 481		3115_3348_4573|0,121132	3115 3348 4573		3135_112_3137|1,31332	3135 112 3137		3135_3968_5057|1,33322	3135 3968 5057		3145_2928_4297|0,123312	3145 2928 4297		315_1972_1997|1,22113	315 1972 1997		315_572_653|0,111131	315 572 653		315_572_653|1,11213	315 572 653		1 315_988_1037|0,12211	315 988 1037		315_988_1037|1,21213	315 988 1037		1 319_360_481|0,1232	319 360 481		319_360_481|1,1122	319 360 481		1 3195_1508_3533|0,133223	3195 1508 3533		321_5720_5729|1,23311	321 5720 5729		3213_6716_7445|0,112221	3213 6716 7445		323_36_325|1,1112	323 36 325		325_228_397|1,2321	325 228 397		3255_3712_4937|1,23322	3255 3712 4937		3283_1044_3445|0,132233	3283 1044 3445		3285_1652_3677|0,112323	3285 1652 3677		329_1080_1129|0,133211	329 1080 1129		329_1080_1129|1,12211	329 1080 1129		1 3293_3276_4645|0,111222	3293 3276 4645		33_544_545|1,1111	33 544 545		33_56_65|0,131	33 56 65		33_56_65|1,211	33 56 65		1 3315_2852_4373|0,123112	3315 2852 4373		333_644_725|0,111121	333 644 725		333_644_725|1,3221	333 644 725		1 3355_348_3373|1,23312	3355 348 3373		3363_116_3365|1,33112	3363 116 3365		3367_3456_4825|0,131122	3367 3456 4825		3367_3456_4825|1,22322	3367 3456 4825		1 3375_7448_8177|0,122321	3375 7448 8177		339_6380_6389|1,23313	339 6380 6389		341_420_541|0,133332	341 420 541		341_420_541|1,2221	341 420 541		1 3431_1560_3769|0,112123	3431 1560 3769		3441_5680_6641|0,132231	3441 5680 6641		345_152_377|0,11123	345 152 377		345_152_377|1,33211	345 152 377		1 3471_3200_4721|0,112312	3471 3200 4721		3471_3200_4721|1,32322	3471 3200 4721		1 35_12_37|0,133	35 12 37		35_12_37|1,12	35 12 37		1 35_612_613|1,1113	35 612 613		351_280_449|1,2122	351 280 449		3515_3828_5197|0,112132	3515 3828 5197		355_2508_2533|1,12313	355 2508 2533		3567_2944_4625|1,32222	3567 2944 4625		357_1276_1325|1,32131	357 1276 1325		357_7076_7085|1,23331	357 7076 7085		357_76_365|0,113333	357 76 365		357_76_365|1,11121	357 76 365		1 3589_4020_5389|0,113232	3589 4020 5389		3599_120_3601|1,33132	3599 120 3601		3627_6364_7325|0,122131	3627 6364 7325		365_2652_2677|1,22131	365 2652 2677		3655_2688_4537|1,22222	3655 2688 4537		3683_7644_8485|0,123221	3683 7644 8485		369_800_881|0,11321	369 800 881		37_684_685|1,1131	37 684 685		3705_3752_5273|0,133122	3705 3752 5273		371_1380_1429|0,123111	371 1380 1429		3731_3300_4981|0,113212	3731 3300 4981		3735_2432_4457|0,122113	3735 2432 4457		3735_2432_4457|1,23222	3735 2432 4457		1 3737_4416_5785|0,122332	3737 4416 5785		3741_7900_8741|0,121221	3741 7900 8741		375_7808_7817|1,23333	375 7808 7817		377_336_505|0,1212	377 336 505		3807_2176_4385|0,132313	3807 2176 4385		3807_2176_4385|1,33222	3807 2176 4385		1 3813_3484_5165|0,131312	3813 3484 5165		3843_124_3845|1,33312	3843 124 3845		387_884_965|0,133321	387 884 965		3871_1920_4321|0,131323	3871 1920 4321		3871_1920_4321|1,33232	3871 1920 4321		1 39_760_761|1,1133	39 760 761		39_80_89|0,121	39 80 89		39_80_89|1,23	39 80 89		1 3901_2340_4549|0,131213	3901 2340 4549		391_120_409|0,11233	391 120 409		391_120_409|1,2132	391 120 409		1 3927_1664_4265|1,23232	3927 1664 4265		3975_1408_4217|1,22232	3975 1408 4217		3977_3864_5545|0,123322	3977 3864 5545		399_1600_1649|1,3233	399 1600 1649		399_40_401|1,1132	399 40 401		4015_1152_4177|1,32232	4015 1152 4177		4017_4544_6065|0,131232	4017 4544 6065		4029_1820_4421|0,132123	4029 1820 4421		403_396_565|0,1322	403 396 565		403_396_565|1,11212	403 396 565		1 4047_896_4145|1,32332	4047 896 4145		405_3268_3293|1,12331	405 3268 3293		4059_4060_5741|0,12222	4059 4060 5741		4061_8100_9061|0,122121	4061 8100 9061		407_624_745|0,12331	407 624 745		407_624_745|1,1323	407 624 745		1 4071_640_4121|1,22332	4071 640 4121		4087_384_4105|1,23332	4087 384 4105		4095_128_4097|1,33332	4095 128 4097		41_840_841|1,1311	41 840 841		413_1716_1765|0,122111	413 1716 1765		413_1716_1765|1,21231	413 1716 1765		1 4141_4260_5941|0,121122	4141 4260 5941		415_3432_3457|1,22133	415 3432 3457		4165_2412_4813|0,121313	4165 2412 4813		423_1064_1145|1,12123	423 1064 1145		4247_8904_9865|0,132221	4247 8904 9865		425_168_457|0,111133	425 168 457		4263_2584_4985|0,112213	4263 2584 4985		427_1836_1885|1,12213	427 1836 1885		429_460_629|0,11132	429 460 629		429_700_821|0,11231	429 700 821		429_700_821|1,32121	429 700 821		1 43_924_925|1,1313	43 924 925		4305_4672_6353|0,132132	4305 4672 6353		4345_1608_4633|0,122133	4345 1608 4633		435_308_533|1,32112	435 308 533		4365_3692_5717|0,122112	4365 3692 5717		437_84_445|0,123333	437 84 445		437_84_445|1,11321	437 84 445		1 441_1160_1241|0,113311	441 1160 1241		45_1012_1013|1,1331	45 1012 1013		45_28_53|0,113	45 28 53		45_28_53|1,121	45 28 53		1 451_780_901|0,13131	451 780 901		451_780_901|1,13213	451 780 901		1 4515_4588_6437|0,113122	4515 4588 6437		455_2088_2137|1,32133	455 2088 2137		455_4128_4153|1,12333	455 4128 4153		455_528_697|0,11332	455 528 697		455_528_697|1,1322	455 528 697		1 459_220_509|0,1223	459 220 509		459_220_509|1,2212	459 220 509		1 4641_2840_5441|0,123213	4641 2840 5441		465_4312_4337|1,22311	465 4312 4337		47_1104_1105|1,1333	47 1104 1105		475_132_493|0,111333	475 132 493		475_132_493|1,21112	475 132 493		1 477_1364_1445|1,11231	477 1364 1445		4773_5236_7085|0,123132	4773 5236 7085		4785_2272_5297|0,113223	4785 2272 5297		481_600_769|1,23211	481 600 769		4815_4712_6737|0,112322	4815 4712 6737		483_44_485|1,1312	483 44 485		4859_5460_7309|0,112232	4859 5460 7309		4895_2448_5473|0,132323	4895 2448 5473		49_1200_1201|1,3111	49 1200 1201		4905_4928_6953|0,133222	4905 4928 6953		493_276_565|0,13313	493 276 565		493_276_565|1,21321	493 276 565		1 4935_4408_6617|0,131212	4935 4408 6617		495_1472_1553|1,3223	495 1472 1553		495_952_1073|1,21123	495 952 1073		5_12_13|0,11	5 12 13		5_12_13|1,1	5 12 13		1 5029_4620_6829|0,132312	5029 4620 6829		5073_2336_5585|0,123123	5073 2336 5585		51_1300_1301|1,3113	51 1300 1301		51_140_149|0,1311	51 140 149		51_140_149|1,213	51 140 149		1 511_2640_2689|1,21233	511 2640 2689		513_184_545|0,13133	513 184 545		515_5292_5317|1,22313	515 5292 5317		517_1044_1165|0,13121	517 1044 1165		517_1044_1165|1,11221	517 1044 1165		1 525_2788_2837|1,12231	525 2788 2837		525_92_533|1,13121	525 92 533		527_336_625|0,13113	527 336 625		527_336_625|1,1222	527 336 625		1 53_1404_1405|1,3131	53 1404 1405		531_1700_1781|0,113211	531 1700 1781		533_756_925|1,21221	533 756 925		5335_3192_6217|0,121213	5335 3192 6217		5341_4740_7141|0,112212	5341 4740 7141		5355_6068_8093|0,121232	5355 6068 8093		539_1140_1261|0,11221	539 1140 1261		5405_5508_7717|0,112122	5405 5508 7717		5439_4960_7361|0,121312	5439 4960 7361		55_1512_1513|1,3133	55 1512 1513		55_48_73|0,112	55 48 73		55_48_73|1,22	55 48 73		1 551_240_601|0,111123	551 240 601		551_240_601|1,1232	551 240 601		1 553_3096_3145|1,32311	553 3096 3145		555_572_797|0,11122	555 572 797		555_572_797|1,13212	555 572 797		1 559_840_1009|0,113331	559 840 1009		559_840_1009|1,3123	559 840 1009		1 5605_2772_6253|0,121323	5605 2772 6253		561_1240_1361|0,12321	561 1240 1361		561_1240_1361|1,22211	561 1240 1361		1 565_6372_6397|1,22331	565 6372 6397		57_1624_1625|1,3311	57 1624 1625		57_176_185|0,1211	57 176 185		57_176_185|1,2111	57 176 185		1 5733_5644_8045|0,131322	5733 5644 8045		575_48_577|1,1332	575 48 577		5757_3476_6725|0,132213	5757 3476 6725		5763_5084_7685|0,123212	5763 5084 7685		5781_6460_8669|0,123232	5781 6460 8669		5785_1848_6073|0,122233	5785 1848 6073		583_1344_1465|1,2223	583 1344 1465		585_2072_2153|1,32211	585 2072 2153		585_928_1097|0,111331	585 928 1097		589_300_661|0,13323	589 300 661		59_1740_1741|1,3313	59 1740 1741		5917_2844_6565|0,131223	5917 2844 6565		6027_9964_11645|0,122231	6027 9964 11645		609_200_641|0,13233	609 200 641		61_1860_1861|1,3331	61 1860 1861		611_1020_1189|0,13231	611 1020 1189		611_1020_1189|1,31213	611 1020 1189		1 615_728_953|0,12332	615 728 953		615_728_953|1,3122	615 728 953		1 615_7552_7577|1,22333	615 7552 7577		621_100_629|1,13321	621 100 629		623_3936_3985|1,12233	623 3936 3985		627_364_725|0,11313	627 364 725		627_364_725|1,21212	627 364 725		1 629_540_829|0,13112	629 540 829		629_540_829|1,22121	629 540 829		1 63_16_65|0,1333	63 16 65		63_16_65|1,32	63 16 65		1 63_1984_1985|1,3333	63 1984 1985		6321_7120_9521|0,132232	6321 7120 9521		637_1116_1285|0,12131	637 1116 1285		6375_6512_9113|0,132122	6375 6512 9113		639_2480_2561|1,11233	639 2480 2561		6405_3652_7373|0,122313	6405 3652 7373		649_1680_1801|0,123311	649 1680 1801		65_2112_2113|1,11111	65 2112 2113		65_72_97|0,132	65 72 97		65_72_97|1,1211	65 72 97		1 651_4300_4349|1,32313	651 4300 4349		663_1216_1385|1,2323	663 1216 1385		663_616_905|0,13312	663 616 905		663_616_905|1,21122	663 616 905		1 665_432_793|0,12113	665 432 793		667_156_685|0,112333	667 156 685		667_156_685|1,2312	667 156 685		1 6695_3192_7417|0,112223	6695 3192 7417		67_2244_2245|1,11113	67 2244 2245		671_1800_1921|0,112311	671 1800 1921		671_1800_1921|1,32123	671 1800 1921		1 675_52_677|1,3112	675 52 677		6765_6052_9077|0,121212	6765 6052 9077		69_2380_2381|1,11131	69 2380 2381		69_260_269|0,13111	69 260 269		69_260_269|1,231	69 260 269		1 693_1924_2045|0,131311	693 1924 2045		693_1924_2045|1,13231	693 1924 2045		1 697_696_985|0,1222	697 696 985		7_24_25|0,111	7 24 25		7_24_25|1,3	7 24 25		1 703_504_865|1,12122	703 504 865		71_2520_2521|1,11133	71 2520 2521		713_216_745|0,111233	713 216 745		715_1428_1597|0,12121	715 1428 1597		7205_7068_10093|0,132322	7205 7068 10093		7239_6440_9689|0,132212	7239 6440 9689		725_108_733|1,31121	725 108 733		73_2664_2665|1,11311	73 2664 2665		731_780_1069|0,111132	731 780 1069		731_780_1069|1,31212	731 780 1069		1 7315_7332_10357|0,113222	7315 7332 10357		735_1088_1313|0,123331	735 1088 1313		735_1088_1313|1,3323	735 1088 1313		1 7383_3344_8105|0,122123	7383 3344 8105		741_1540_1709|0,13221	741 1540 1709		741_1540_1709|1,13221	741 1540 1709		1 741_580_941|1,12321	741 580 941		7421_15540_17221|0,122221	7421 15540 17221		747_3404_3485|1,32213	747 3404 3485		749_5700_5749|1,32331	749 5700 5749		75_2812_2813|1,11313	75 2812 2813		75_308_317|0,12111	75 308 317		75_308_317|1,2113	75 308 317		1 7521_3560_8321|0,123223	7521 3560 8321		759_2320_2441|0,131211	759 2320 2441		759_2320_2441|1,11223	759 2320 2441		1 759_280_809|0,12133	759 280 809		759_280_809|1,12132	759 280 809		1 765_868_1157|0,11232	765 868 1157		765_868_1157|1,12221	765 868 1157		1 767_1656_1825|0,111321	767 1656 1825		77_2964_2965|1,11331	77 2964 2965		77_36_85|0,123	77 36 85		77_36_85|1,321	77 36 85		1 775_168_793|1,21132	775 168 793		7755_8428_11453|0,122132	7755 8428 11453		777_464_905|0,11213	777 464 905		779_660_1021|0,12112	779 660 1021		781_2460_2581|0,112211	781 2460 2581		783_56_785|1,3132	783 56 785		79_3120_3121|1,11333	79 3120 3121		7923_8036_11285|0,123122	7923 8036 11285		793_1776_1945|0,113321	793 1776 1945		795_1292_1517|0,111231	795 1292 1517		795_1292_1517|1,33213	795 1292 1517		1 799_960_1249|0,113332	799 960 1249		799_960_1249|1,3322	799 960 1249		1 8023_3864_8905|0,121223	8023 3864 8905		803_2604_2725|0,123211	803 2604 2725		803_2604_2725|1,22213	803 2604 2725		1 805_348_877|1,23121	805 348 877		81_3280_3281|1,13111	81 3280 3281		8165_4092_9133|0,122323	8165 4092 9133		817_744_1105|0,11312	817 744 1105		819_1900_2069|1,23213	819 1900 2069		83_3444_3445|1,13113	83 3444 3445		8319_8200_11681|0,121322	8319 8200 11681		837_116_845|1,31321	837 116 845		847_7296_7345|1,32333	847 7296 7345		8479_7800_11521|0,122312	8479 7800 11521		85_132_157|0,1331	85 132 157		85_132_157|1,221	85 132 157		1 85_3612_3613|1,13131	85 3612 3613		851_420_949|0,11323	851 420 949		855_832_1193|0,13322	855 832 1193		855_832_1193|1,2322	855 832 1193		1 87_3784_3785|1,13133	87 3784 3785		87_416_425|0,131111	87 416 425		87_416_425|1,233	87 416 425		1 871_2160_2329|1,21223	871 2160 2329		89_3960_3961|1,13311	89 3960 3961		8925_4268_9893|0,132223	8925 4268 9893		893_924_1285|0,111122	893 924 1285		8967_8944_12665|0,131222	8967 8944 12665		897_496_1025|0,133313	897 496 1025		899_60_901|1,3312	899 60 901		9_40_41|0,1111	9 40 41		9_40_41|1,11	9 40 41		1 903_704_1145|1,2222	903 704 1145		909_5060_5141|1,32231	909 5060 5141		91_4140_4141|1,13313	91 4140 4141		91_60_109|0,1113	91 60 109		91_60_109|1,212	91 60 109		1 93_4324_4325|1,13331	93 4324 4325		93_476_485|0,121111	93 476 485		93_476_485|1,2131	93 476 485		1 931_1020_1381|0,13132	931 1020 1381		931_1020_1381|1,33212	931 1020 1381		1 935_1368_1657|1,11123	935 1368 1657		935_3552_3673|1,13233	935 3552 3673		943_576_1105|0,13213	943 576 1105		943_576_1105|1,3222	943 576 1105		1 945_248_977|0,131333	945 248 977		949_2580_2749|0,132311	949 2580 2749		949_2580_2749|1,31231	949 2580 2749		1 95_168_193|0,1131	95 168 193		95_168_193|1,123	95 168 193		1 95_4512_4513|1,13333	95 4512 4513		957_124_965|1,33121	957 124 965		969_1120_1481|0,111332	969 1120 1481		969_1480_1769|0,112331	969 1480 1769		97_4704_4705|1,31111	97 4704 4705		975_2728_2897|0,121311	975 2728 2897		975_448_1073|0,13123	975 448 1073		975_448_1073|1,3232	975 448 1073		1 987_884_1325|0,11212	987 884 1325		989_660_1189|0,131113	989 660 1189		99_20_101|0,13333	99 20 101		99_20_101|1,112	99 20 101		1 99_4900_4901|1,31113	99 4900 4901		9975_6032_11657|0,122213	9975 6032 11657		999_320_1049|0,12233	999 320 1049		999_320_1049|1,2232	999 320 1049		1


 * added triplets contained in both trees; regards Gangleri

details to Price's tree
to be continued 2A02:810D:41C0:134:6C5D:2ACF:DA6:D533 (talk) 18:07, 12 July 2019 (UTC)

0 1 4 4 12 9 40 25 144 81 544 289 2112 1089 68 1156 2244 1156 36 324 76 361 1224 1296 1368 1296 612 324 2380 1225 72 1296 2520 1296 20 100 44 121 104 169 396 484 572 484 360 400 1044 841 880 1600 2320 1600 440 400 1364 961 720 1600 2480 1600 180 100 684 361 2664 1369 76 1444 2812 1444 40 400 84 441 1520 1600 1680 1600 760 400 2964 1521 80 1600 3120 1600 12 36 28 49 72 81 208 169 180 324 468 324 140 196 380 361 504 784 1064 784 252 196 828 529 280 784 1288 784 120 144 340 289 1080 729 476 1156 1836 1156 336 576 868 961 1632 2304 2976 2304 816 576 2788 1681 672 2304 3936 2304 168 144 532 361 1848 1089 380 1444 2508 1444 240 576 580 841 1824 2304 2784 2304 912 576 3268 1849 480 2304 4128 2304 60 36 220 121 840 441 3280 1681 84 1764 3444 1764 44 484 92 529 1848 1936 2024 1936 924 484 3612 1849 88 1936 3784 1936 24 144 52 169 120 225 572 676 780 676 528 576 1540 1225 1248 2304 3360 2304 624 576 1924 1369 1056 2304 3552 2304 264 144 1012 529 3960 2025 92 2116 4140 2116 48 576 100 625 2208 2304 2400 2304 1104 576 4324 2209 96 2304 4512 2304 8 16 20 25 56 49 176 121 608 361 132 484 836 484 84 196 204 289 616 784 952 784 308 196 1100 625 168 784 1400 784 60 100 156 169 456 361 364 676 988 676 280 400 756 729 1040 1600 2160 1600 520 400 1716 1089 560 1600 2640 1600 140 100 476 289 1736 961 204 1156 2108 1156 120 400 276 529 1360 1600 1840 1600 680 400 2516 1369 240 1600 2960 1600 48 64 132 121 408 289 1392 841 340 1156 1972 1156 220 484 540 729 1496 1936 2376 1936 748 484 2652 1521 440 1936 3432 1936 160 256 420 441 1240 961 924 1764 2604 1764 704 1024 1892 1849 2688 4096 5504 4096 1344 1024 4452 2809 1408 4096 6784 4096 352 256 1188 729 4312 2401 540 2916 5292 2916 320 1024 740 1369 3456 4096 4736 4096 1728 1024 6372 3481 640 4096 7552 4096 80 64 260 169 920 529 3440 1849 276 2116 3956 2116 156 676 348 841 2392 2704 3016 2704 1196 676 4508 2401 312 2704 5096 2704 96 256 228 361 600 625 988 1444 1900 1444 832 1024 2340 2025 2432 4096 5760 4096 1216 1024 3876 2601 1664 4096 6528 4096 416 256 1508 841 5720 3025 348 3364 6380 3364 192 1024 420 1225 3712 4096 4480 4096 1856 1024 7076 3721 384 4096 7808 4096 24 16 84 49 312 169 1200 625 4704 2401 100 2500 4900 2500 52 676 108 729 2600 2704 2808 2704 1300 676 5100 2601 104 2704 5304 2704 28 196 60 225 136 289 780 900 1020 900 728 784 2132 1681 1680 3136 4592 3136 840 784 2580 1849 1456 3136 4816 3136 364 196 1404 729 5512 2809 108 2916 5724 2916 56 784 116 841 3024 3136 3248 3136 1512 784 5940 3025 112 3136 6160 3136 16 64 36 81 88 121 240 225 308 484 660 484 252 324 700 625 792 1296 1800 1296 396 324 1276 841 504 1296 2088 1296 224 256 644 529 2072 1369 828 2116 3404 2116 576 1024 1476 1681 2944 4096 5248 4096 1472 1024 5060 3025 1152 4096 7040 4096 288 256 900 625 3096 1849 700 2500 4300 2500 448 1024 1092 1521 3200 4096 4992 4096 1600 1024 5700 3249 896 4096 7296 4096 112 64 420 225 1624 841 6384 3249 116 3364 6612 3364 60 900 124 961 3480 3600 3720 3600 1740 900 6844 3481 120 3600 7080 3600 32 256 68 289 152 361 1020 1156 1292 1156 960 1024 2820 2209 2176 4096 6016 4096 1088 1024 3332 2401 1920 4096 6272 4096 480 256 1860 961 7320 3721 124 3844 7564 3844 64 1024 132 1089 3968 4096 4224 4096 1984 1024 7812 3969 128 4096 8064 4096


 * added sorted (b, n) pairs for the Price's tree; regards Gangleri 217.86.249.144 (talk) 14:34, 16 July 2019 (UTC)
 * updated sorted properly by path no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 12:27, 19 July 2019 (UTC)
 * getting harassed by a nasty excell sort bug no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 13:16, 19 July 2019 (UTC)
 * now replaced with values for the Price tree no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 13:28, 19 July 2019 (UTC)
 * getting harassed by a nasty excell sort bug; using fake number workaround no bias — קיין אומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contribs 13:50, 19 July 2019 (UTC)

comments
from user:לערי ריינהארט/sandbox; regards Gangleri 2A02:810D:41C0:134:6C5D:2ACF:DA6:D533 (talk) 18:07, 12 July 2019 (UTC)
 * like Cronus, Wikipedia devours its children
 * regards Gangleri 217.86.249.144 (talk) 15:09, 16 July 2019 (UTC)

to do
consider:
 * rational triangles a redirect to integer triangles;
 * rational trigonometry not considered here further
 * triangles where all trigonometric values of the three angles see also trigonometric functions: sine, cosine, tangent, and their reciprocals cosecant, secant and cotangent are rational numbers; these are Pythagorean triangles redirecting to Pythagorean triple.

the six values for the trigonometric functions: sine, cosine, tangent, and their reciprocals cosecant, secant and cotangent are keys (cryptography)  each enabeling to identify a unique primitive Pythagorean triple today a redirect to Pythagorean triple; a better redirect could be tree of primitive Pythagorean triples; the later should be moved to trees of primitive Pythagorean triples|undefined; the representation of the rational number values associated with each trigonometric function would be the pair of two integer numbers used to codify the rational number as a fraction

So far 13 keys for each primitive Pythagorean triple are identified. It is a bias issue of each contributor how many and which to be used. regards Gangleri 95.91.248.133 (talk) 10:29, 17 July 2019 (UTC)
 * Pythagorean numbers a redirect to Pythagorean prime
 * Pythagorean pair / Pythagorean pairs referred in file:Pythagorean pairs.svg

regards no bias — קיין ומוויסנדיק פּרעפֿערענצן — keyn umvisndik preferentsn talk contributions 20:47, 17 July 2019 (UTC)