Pythagorean triples – Equal products of two legs

 
 
The following integers

a_1 \; = \; p^2 \; - \; q^2
b_1 \; = \; 2 \, p \, q
c_1 \; = \; p^2 \; + \; q^2

a_2 \; = \; r^2 \; - \; s^2
b_2 \; = \; 2 \, r \, s
c_2 \; = \; r^2 \; + \; s^2

define two right triangles   (a_1, \; b_1, \; c_1),   (a_2, \; b_2, \; c_2)

 

Can you find a pair of Pythagorean triples with equal products of two legs:

(2 \,p \,q) \,(p^2 - q^2) \; = \; (2 \,r \,s) \,(r^2 - s^2)

 
 

Related blog:
  Pythagorean triples – Equal products of a leg and hypotenuse

 
 
For example,

(24,   70,   74)   =   2 × [12, 35, 37]
(40,   42,   58)   =   2 × [20, 21, 29]
(15,   112,   113)   is a primitive

24 \times 70 \; = \; 40 \times 42 \; = \; 15 \times 112 \; = \; 1680

(360,   378,   522)   =   18 × [20, 21, 29]
(216,   630,   666)   =   18 × [12, 35, 37]
(135,   1008,   1017)   =   9 × [15, 112, 113]

360 \times 378 \; = \; 216 \times 630 \; = \; 135 \times 1008 \; = \; 136080

(280,   2442,   2458)   =   2 × [140, 1221, 1229]
(518,   1320,   1418)   =   2 × [259, 660, 709]
(111,   6160,   6161)   is a primitive
(231,   2960,   2969)   is a primitive

280 \times 2442 = 518 \times 1320 = 111 \times 6160 = 231 \times 2960 = 683760

(1118,   1680,   2018)   =   2 × [559, 840, 1009]
(624,   3010,   3074)   =   2 × [312, 1505, 1537]
(455,   4128,   4153)   is a primitive

1118 \times 1680 \; = \; 624 \times 3010 \; = \; 455 \times 4128 \; = \; 1878240

(1920,   2912,   3488)   =   32 × [60, 91, 109]
(1680,   3328,   3728)   =   16 × [105, 208, 233]
(896,   6240,   6304)   =   32 × [28, 195, 197]

1920 \times 2912 \; = \; 1680 \times 3328 \; = \; 896 \times 6240 \; = \; 5591040

(2482,   5040,   5618)   =   2 × [1241, 2520, 2809]
(1360,   9198,   9298)   =   2 × [680, 4599, 4649]
(1071,   11680,   11729)   is a primitive

2482 \times 5040 = 1360 \times 9198 = 1071 \times 11680 \; = \; 12509280

(504,   7930,   7946)   =   2 × [252, 3965, 3973]
(1098,   3640,   3802)   =   2 × [549, 1820, 1901]
(585,   6832,   6857)   is a primitive

504 \times 7930 = 1098 \times 3640 = 585 \times 6832 = 3996720

(4662,   11880,   12762)   =   18 × [259, 660, 709]
(2079,   26640,   26721)   =   9 × [231, 2960, 2969]
(2520,   21978,   22122)   =   18 × [140, 1221, 1229
(999,   55440,   55449)   =   9 × [111, 6160, 6161]

4662 \times 11880 = 2079 \times 26640 = 2520 \times 21978 = 999 \times 55440 = 55384560

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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One Response to Pythagorean triples – Equal products of two legs

  1. paul says:

    Here are those with a <=1000, format is:-
    {{a1, b1, c1, a1 x b1}, {a2, b2, c2, a2 x b2}, ….}

    {{15,112,113,1680},{24,70,74,1680},{40,42,58,1680}}
    {{22,120,122,2640},{48,55,73,2640}}
    {{27,364,365,9828},{39,252,255,9828}}
    {{28,195,197,5460},{60,91,109,5460}}
    {{30,224,226,6720},{48,140,148,6720},{80,84,116,6720}}
    {{36,105,111,3780},{60,63,87,3780}}
    {{40,399,401,15960},{95,168,193,15960}}
    {{44,240,244,10560},{96,110,146,10560}}
    {{45,336,339,15120},{72,210,222,15120},{120,126,174,15120}}
    {{49,1200,1201,58800},{210,280,350,58800}}
    {{54,728,730,39312},{78,504,510,39312}}
    {{55,1512,1513,83160},{264,315,411,83160}}
    {{56,390,394,21840},{105,208,233,21840},{120,182,218,21840}}
    {{60,175,185,10500},{100,105,145,10500}}
    {{60,448,452,26880},{96,280,296,26880},{160,168,232,26880}}
    {{65,2112,2113,137280},{208,660,692,137280}}
    {{66,360,366,23760},{144,165,219,23760}}
    {{75,560,565,42000},{120,350,370,42000},{200,210,290,42000}}
    {{77,2964,2965,228228},{364,627,725,228228}}
    {{80,798,802,63840},{105,608,617,63840},{190,336,386,63840}}
    {{81,1092,1095,88452},{117,756,765,88452}}
    {{84,245,259,20580},{140,147,203,20580}}
    {{84,585,591,49140},{180,273,327,49140}}
    {{85,720,725,61200},{225,272,353,61200}}
    {{88,480,488,42240},{192,220,292,42240}}
    {{90,672,678,60480},{144,420,444,60480},{240,252,348,60480}}
    {{98,2400,2402,235200},{420,560,700,235200}}
    {{105,784,791,82320},{168,490,518,82320},{280,294,406,82320}}
    {{108,315,333,34020},{180,189,261,34020}}
    {{108,1456,1460,157248},{156,1008,1020,157248}}
    {{110,600,610,66000},{240,275,365,66000}}
    {{110,3024,3026,332640},{480,693,843,332640},{528,630,822,332640}}
    {{111,6160,6161,683760},{231,2960,2969,683760},{280,2442,2458,683760},{518,1320,1418,683760}}
    {{112,780,788,87360},{210,416,466,87360},{240,364,436,87360}}
    {{112,3135,3137,351120},{330,1064,1114,351120}}
    {{116,1680,1684,194880},{192,1015,1033,194880}}
    {{120,896,904,107520},{192,560,592,107520},{320,336,464,107520}}
    {{120,1197,1203,143640},{285,504,579,143640}}
    {{130,4224,4226,549120},{416,1320,1384,549120}}
    {{132,385,407,50820},{220,231,319,50820}}
    {{132,720,732,95040},{288,330,438,95040}}
    {{132,1085,1093,143220},{341,420,541,143220}}
    {{135,1008,1017,136080},{216,630,666,136080},{360,378,522,136080}}
    {{135,1820,1825,245700},{195,1260,1275,245700}}
    {{140,975,985,136500},{300,455,545,136500}}
    {{140,1221,1229,170940},{259,660,709,170940}}
    {{144,1725,1731,248400},{200,1242,1258,248400}}
    {{147,3600,3603,529200},{630,840,1050,529200}}
    {{150,1120,1130,168000},{240,700,740,168000},{400,420,580,168000}}
    {{153,420,447,64260},{189,340,389,64260}}
    {{154,840,854,129360},{336,385,511,129360}}
    {{154,5928,5930,912912},{728,1254,1450,912912}}
    {{156,455,481,70980},{260,273,377,70980}}
    {{160,231,281,36960},{176,210,274,36960}}
    {{160,1596,1604,255360},{210,1216,1234,255360},{380,672,772,255360}}
    {{162,2184,2190,353808},{234,1512,1530,353808}}
    {{165,1232,1243,203280},{264,770,814,203280},{440,462,638,203280}}
    {{165,4536,4539,748440},{792,945,1233,748440}}
    {{168,1170,1182,196560},{315,624,699,196560},{360,546,654,196560}}
    {{170,1440,1450,244800},{450,544,706,244800}}
    {{170,7224,7226,1228080},{688,1785,1913,1228080}}
    {{171,528,555,90288},{297,304,425,90288}}
    {{172,3696,3700,635712},{704,903,1145,635712}}
    {{176,960,976,168960},{384,440,584,168960}}
    {{180,385,425,69300},{252,275,373,69300}}
    {{180,525,555,94500},{300,315,435,94500}}
    {{180,1344,1356,241920},{288,840,888,241920},{480,504,696,241920}}
    {{180,4048,4052,728640},{495,1472,1553,728640}}
    {{186,952,970,177072},{336,527,625,177072}}
    {{189,2548,2555,481572},{273,1764,1785,481572}}
    {{195,1456,1469,283920},{312,910,962,283920},{520,546,754,283920}}
    {{195,6336,6339,1235520},{624,1980,2076,1235520}}
    {{196,1365,1379,267540},{420,637,763,267540}}
    {{196,4800,4804,940800},{840,1120,1400,940800}}
    {{198,1080,1098,213840},{432,495,657,213840}}
    {{200,1995,2005,399000},{475,840,965,399000}}
    {{204,595,629,121380},{340,357,493,121380}}
    {{210,1568,1582,329280},{336,980,1036,329280},{560,588,812,329280}}
    {{216,2912,2920,628992},{312,2016,2040,628992}}
    {{220,459,509,100980},{255,396,471,100980}}
    {{220,1200,1220,264000},{480,550,730,264000}}
    {{220,6048,6052,1330560},{960,1386,1686,1330560}}
    {{222,12320,12322,2735040},{462,5920,5938,2735040},{560,4884,4916,2735040}}
    {{224,1560,1576,349440},{420,832,932,349440},{480,728,872,349440}}
    {{224,6270,6274,1404480},{660,2128,2228,1404480}}
    {{225,1680,1695,378000},{360,1050,1110,378000},{600,630,870,378000}}
    {{228,665,703,151620},{380,399,551,151620}}
    {{231,1260,1281,291060},{245,1188,1213,291060}}
    {{231,3808,3815,879648},{833,1056,1345,879648}}
    {{232,3360,3368,779520},{384,2030,2066,779520}}
    {{240,1792,1808,430080},{384,1120,1184,430080},{640,672,928,430080}}
    {{240,2394,2406,574560},{315,1824,1851,574560},{570,1008,1158,574560}}
    {{242,1320,1342,319440},{528,605,803,319440}}
    {{243,3276,3285,796068},{351,2268,2295,796068}}
    {{252,735,777,185220},{420,441,609,185220}}
    {{252,1755,1773,442260},{540,819,981,442260}}
    {{252,3965,3973,999180},{549,1820,1901,999180}}
    {{255,1904,1921,485520},{408,1190,1258,485520},{680,714,986,485520}}
    {{255,2160,2175,550800},{675,816,1059,550800}}
    {{260,1680,1700,436800},{448,975,1073,436800}}
    {{260,8448,8452,2196480},{832,2640,2768,2196480}}
    {{264,950,986,250800},{285,880,925,250800}}
    {{264,1440,1464,380160},{576,660,876,380160}}
    {{264,2170,2186,572880},{385,1488,1537,572880},{682,840,1082,572880}}
    {{270,2016,2034,544320},{432,1260,1332,544320},{720,756,1044,544320}}
    {{270,3640,3650,982800},{390,2520,2550,982800}}
    {{276,805,851,222180},{460,483,667,222180}}
    {{280,759,809,212520},{385,552,673,212520}}
    {{280,1950,1970,546000},{525,1040,1165,546000},{600,910,1090,546000}}
    {{280,2793,2807,782040},{665,1176,1351,782040}}
    {{280,3915,3925,1096200},{600,1827,1923,1096200}}
    {{285,2128,2147,606480},{456,1330,1406,606480},{760,798,1102,606480}}
    {{286,1560,1586,446160},{624,715,949,446160}}
    {{288,3450,3462,993600},{400,2484,2516,993600}}
    {{297,4004,4015,1189188},{429,2772,2805,1189188}}
    {{300,875,925,262500},{500,525,725,262500}}
    {{300,2240,2260,672000},{480,1400,1480,672000},{800,840,1160,672000}}
    {{306,840,894,257040},{378,680,778,257040}}
    {{308,1680,1708,517440},{672,770,1022,517440}}
    {{308,2145,2167,660660},{660,1001,1199,660660}}
    {{312,1505,1537,469560},{559,840,1009,469560}}
    {{312,2695,2713,840840},{440,1911,1961,840840}}
    {{315,2352,2373,740880},{504,1470,1554,740880},{840,882,1218,740880}}
    {{320,462,562,147840},{352,420,548,147840}}
    {{320,3192,3208,1021440},{420,2432,2468,1021440},{760,1344,1544,1021440}}
    {{324,945,999,306180},{540,567,783,306180}}
    {{324,4368,4380,1415232},{468,3024,3060,1415232}}
    {{330,1800,1830,594000},{720,825,1095,594000}}
    {{330,2464,2486,813120},{528,1540,1628,813120},{880,924,1276,813120}}
    {{333,18480,18483,6153840},{693,8880,8907,6153840},{840,7326,7374,6153840}}
    {{336,2340,2364,786240},{630,1248,1398,786240},{720,1092,1308,786240}}
    {{336,9405,9411,3160080},{990,3192,3342,3160080}}
    {{340,2880,2900,979200},{900,1088,1412,979200}}
    {{342,1056,1110,361152},{594,608,850,361152}}
    {{345,2576,2599,888720},{552,1610,1702,888720},{920,966,1334,888720}}
    {{348,1015,1073,353220},{580,609,841,353220}}
    {{348,5040,5052,1753920},{576,3045,3099,1753920}}
    {{351,4732,4745,1660932},{507,3276,3315,1660932}}
    {{352,1920,1952,675840},{768,880,1168,675840}}
    {{360,770,850,277200},{504,550,746,277200}}
    {{360,2688,2712,967680},{576,1680,1776,967680},{960,1008,1392,967680}}
    {{360,3591,3609,1292760},{855,1512,1737,1292760}}
    {{360,8096,8104,2914560},{990,2944,3106,2914560}}
    {{364,2535,2561,922740},{780,1183,1417,922740}}
    {{372,1085,1147,403620},{620,651,899,403620}}
    {{372,1904,1940,708288},{672,1054,1250,708288}}
    {{374,2040,2074,762960},{816,935,1241,762960}}
    {{375,2800,2825,1050000},{600,1750,1850,1050000},{1000,1050,1450,1050000}}
    {{378,5096,5110,1926288},{546,3528,3570,1926288}}
    {{390,2912,2938,1135680},{624,1820,1924,1135680}}
    {{392,2730,2758,1070160},{735,1456,1631,1070160},{840,1274,1526,1070160}}
    {{396,1155,1221,457380},{660,693,957,457380}}
    {{396,2160,2196,855360},{864,990,1314,855360}}
    {{400,3990,4010,1596000},{525,3040,3085,1596000},{950,1680,1930,1596000}}
    {{405,3024,3051,1224720},{648,1890,1998,1224720}}
    {{405,5460,5475,2211300},{585,3780,3825,2211300}}
    {{418,2280,2318,953040},{912,1045,1387,953040}}
    {{420,1225,1295,514500},{700,735,1015,514500}}
    {{420,2925,2955,1228500},{900,1365,1635,1228500}}
    {{420,3136,3164,1317120},{672,1960,2072,1317120}}
    {{420,3663,3687,1538460},{777,1980,2127,1538460}}
    {{432,5175,5193,2235600},{600,3726,3774,2235600}}
    {{432,5824,5840,2515968},{624,4032,4080,2515968}}
    {{435,3248,3277,1412880},{696,2030,2146,1412880}}
    {{440,918,1018,403920},{510,792,942,403920}}
    {{440,2400,2440,1056000},{960,1100,1460,1056000}}
    {{444,1295,1369,574980},{740,777,1073,574980}}
    {{444,24640,24644,10940160},{924,11840,11876,10940160}}
    {{448,3120,3152,1397760},{840,1664,1864,1397760},{960,1456,1744,1397760}}
    {{450,3360,3390,1512000},{720,2100,2220,1512000}}
    {{455,4128,4153,1878240},{624,3010,3074,1878240}}
    {{459,1260,1341,578340},{567,1020,1167,578340}}
    {{459,6188,6205,2840292},{663,4284,4335,2840292}}
    {{462,2520,2562,1164240},{490,2376,2426,1164240}}
    {{464,6720,6736,3118080},{768,4060,4132,3118080}}
    {{465,3472,3503,1614480},{744,2170,2294,1614480}}
    {{468,1365,1443,638820},{780,819,1131,638820}}
    {{476,1995,2051,949620},{931,1020,1381,949620}}
    {{480,3584,3616,1720320},{768,2240,2368,1720320}}
    {{480,4788,4812,2298240},{630,3648,3702,2298240}}
    {{486,6552,6570,3184272},{702,4536,4590,3184272}}
    {{492,1435,1517,706020},{820,861,1189,706020}}
    {{495,3696,3729,1829520},{792,2310,2442,1829520}}
    {{504,3510,3546,1769040},{945,1872,2097,1769040}}
    {{504,7930,7946,3996720},{585,6832,6857,3996720}}
    {{510,3808,3842,1942080},{816,2380,2516,1942080}}
    {{513,1584,1665,812592},{891,912,1275,812592}}
    {{513,6916,6935,3547908},{741,4788,4845,3547908}}
    {{516,1505,1591,776580},{860,903,1247,776580}}
    {{520,3360,3400,1747200},{896,1950,2146,1747200}}
    {{525,3920,3955,2058000},{840,2450,2590,2058000}}
    {{528,1900,1972,1003200},{570,1760,1850,1003200}}
    {{528,4340,4372,2291520},{770,2976,3074,2291520}}
    {{540,1155,1275,623700},{756,825,1119,623700}}
    {{540,1575,1665,850500},{900,945,1305,850500}}
    {{540,4032,4068,2177280},{864,2520,2664,2177280}}
    {{540,7280,7300,3931200},{780,5040,5100,3931200}}
    {{555,4144,4181,2299920},{888,2590,2738,2299920}}
    {{560,1518,1618,850080},{770,1104,1346,850080}}
    {{560,5586,5614,3128160},{735,4256,4319,3128160}}
    {{564,1645,1739,927780},{940,987,1363,927780}}
    {{567,7644,7665,4334148},{819,5292,5355,4334148}}
    {{570,4256,4294,2425920},{912,2660,2812,2425920}}
    {{576,6900,6924,3974400},{800,4968,5032,3974400}}
    {{580,8400,8420,4872000},{960,5075,5165,4872000}}
    {{585,4368,4407,2555280},{936,2730,2886,2555280}}
    {{588,1715,1813,1008420},{980,1029,1421,1008420}}
    {{594,8008,8030,4756752},{858,5544,5610,4756752}}
    {{600,4480,4520,2688000},{960,2800,2960,2688000}}
    {{612,1680,1788,1028160},{756,1360,1556,1028160}}
    {{615,4592,4633,2824080},{984,2870,3034,2824080}}
    {{621,8372,8395,5199012},{897,5796,5865,5199012}}
    {{624,5390,5426,3363360},{880,3822,3922,3363360}}
    {{640,924,1124,591360},{704,840,1096,591360}}
    {{640,6384,6416,4085760},{840,4864,4936,4085760}}
    {{648,8736,8760,5660928},{936,6048,6120,5660928}}
    {{660,1377,1527,908820},{765,1188,1413,908820}}
    {{675,9100,9125,6142500},{975,6300,6375,6142500}}
    {{693,3780,3843,2619540},{735,3564,3639,2619540}}
    {{720,7182,7218,5171040},{945,5472,5553,5171040}}
    {{720,8625,8655,6210000},{1000,6210,6290,6210000}}
    {{765,2100,2235,1606500},{945,1700,1945,1606500}}
    {{792,2850,2958,2257200},{855,2640,2775,2257200}}
    {{800,1155,1405,924000},{880,1050,1370,924000}}
    {{861,2448,2595,2107728},{952,2214,2410,2107728}}
    {{924,5040,5124,4656960},{980,4752,4852,4656960}}

    Paul.

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