## Equation: a*b*(a^2 – b^2) = x*y*(x^2 – y^2)

Can you find solutions to the equation :

where   $a, \; b, \; x, \; y$   are positive integers

math grad - Interest: Number theory
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### 6 Responses to Equation: a*b*(a^2 – b^2) = x*y*(x^2 – y^2)

1. paul says:

Here are the solutions with b < a <= 100.
Format is {{a, b, a b(a^2 – b^2)}, {x, y, (x^2 – y^2)}}.
where there are more than 2 solutions perm any 2.

{{5,2,210},{6,1,210}}
{{7,3,840},{7,5,840},{8,7,840}}
{{8,3,1320},{11,1,1320}}
{{10,3,2730},{14,1,2730}}
{{10,4,3360},{12,2,3360},{15,1,3360}}
{{11,5,5280},{12,10,5280}}
{{12,7,7980},{20,1,7980}}
{{13,7,10920},{13,8,10920},{15,13,10920}}
{{14,6,13440},{14,10,13440},{16,14,13440}}
{{15,6,17010},{18,3,17010}}
{{15,7,18480},{16,5,18480}}
{{16,6,21120},{22,2,21120}}
{{19,5,31920},{19,16,31920},{21,19,31920}}
{{20,6,43680},{21,5,43680},{28,2,43680}}
{{20,8,53760},{24,4,53760},{30,2,53760}}
{{21,9,68040},{21,15,68040},{24,21,68040}}
{{26,4,68640},{33,32,68640}}
{{21,10,71610},{33,2,71610}}
{{21,11,73920},{22,8,73920}}
{{22,10,84480},{24,20,84480}}
{{22,15,85470},{35,2,85470}}
{{24,7,88536},{31,3,88536}}
{{30,28,97440},{32,3,97440}}
{{23,12,106260},{28,5,106260}}
{{24,9,106920},{33,3,106920}}
{{26,7,114114},{39,38,114114}}
{{24,14,127680},{35,3,127680},{40,2,127680}}
{{25,10,131250},{30,5,131250}}
{{26,14,174720},{26,16,174720},{30,26,174720}}
{{33,5,175560},{56,1,175560}}
{{28,12,215040},{28,20,215040},{32,28,215040}}
{{30,9,221130},{42,3,221130}}
{{28,15,234780},{39,4,234780}}
{{30,12,272160},{36,6,272160},{45,3,272160}}
{{30,22,274560},{65,1,274560}}
{{31,11,286440},{31,24,286440},{35,31,286440}}
{{30,14,295680},{32,10,295680}}
{{32,11,317856},{44,42,317856}}
{{32,12,337920},{44,4,337920}}
{{37,7,341880},{37,33,341880},{40,37,341880},{56,55,341880}}
{{31,17,354144},{34,28,354144}}
{{32,23,364320},{46,44,364320}}
{{35,29,389760},{58,2,389760}}
{{44,5,420420},{52,3,420420}}
{{33,23,425040},{35,11,425040}}
{{33,15,427680},{36,30,427680}}
{{33,19,456456},{77,1,456456}}
{{35,26,499590},{63,2,499590}}
{{35,14,504210},{42,7,504210}}
{{38,10,510720},{38,32,510720},{42,38,510720}}
{{35,15,525000},{35,25,525000},{40,35,525000}}
{{43,8,614040},{85,1,614040}}
{{36,21,646380},{60,3,646380}}
{{40,12,698880},{42,10,698880},{56,4,698880}}
{{38,28,702240},{57,55,702240}}
{{40,15,825000},{55,5,825000}}
{{40,16,860160},{48,8,860160},{60,4,860160}}
{{39,21,884520},{39,24,884520},{45,39,884520}}
{{43,13,939120},{43,35,939120},{48,43,939120}}
{{42,18,1088640},{42,30,1088640},{48,42,1088640}}
{{52,8,1098240},{66,64,1098240}}
{{42,20,1145760},{55,7,1145760},{66,4,1145760}}
{{42,22,1182720},{44,16,1182720}}
{{43,21,1271424},{86,2,1271424}}
{{44,20,1351680},{48,40,1351680}}
{{44,30,1367520},{70,4,1367520},{77,3,1367520}}
{{45,18,1377810},{54,9,1377810}}
{{48,14,1416576},{62,6,1416576}}
{{44,26,1441440},{45,32,1441440}}
{{55,9,1457280},{90,2,1457280}}
{{45,21,1496880},{48,15,1496880}}
{{60,56,1559040},{64,6,1559040}}
{{45,28,1563660},{68,5,1563660}}
{{46,31,1647030},{62,7,1647030}}
{{49,16,1681680},{49,39,1681680},{55,49,1681680}}
{{46,24,1700160},{56,10,1700160}}
{{50,15,1706250},{70,5,1706250}}
{{48,18,1710720},{66,6,1710720}}
{{52,14,1825824},{78,76,1825824}}
{{56,11,1857240},{72,5,1857240}}
{{61,9,1998360},{61,56,1998360},{65,61,1998360}}
{{49,21,2016840},{49,35,2016840},{56,49,2016840}}
{{51,40,2042040},{55,13,2042040},{88,3,2042040}}
{{48,28,2042880},{70,6,2042880},{80,4,2042880}}
{{50,20,2100000},{60,10,2100000},{75,5,2100000}}
{{51,35,2456160},{86,84,2456160}}
{{57,15,2585520},{57,48,2585520},{63,57,2585520}}
{{52,28,2795520},{52,32,2795520},{60,52,2795520}}
{{55,22,3074610},{66,11,3074610}}
{{56,21,3169320},{77,7,3169320}}
{{56,22,3267264},{64,13,3267264}}
{{55,25,3300000},{60,50,3300000}}
{{55,36,3423420},{95,4,3423420}}
{{56,24,3440640},{56,40,3440640},{64,56,3440640}}
{{55,32,3521760},{69,11,3521760}}
{{60,18,3538080},{63,15,3538080},{84,6,3538080}}
{{65,14,3666390},{85,6,3666390}}
{{56,30,3756480},{78,8,3756480},{91,5,3756480}}
{{56,34,3769920},{64,55,3769920}}
{{60,24,4354560},{72,12,4354560},{90,6,4354560}}
{{62,22,4583040},{62,48,4583040},{70,62,4583040}}
{{60,28,4730880},{64,20,4730880}}
{{60,35,4987500},{100,5,4987500}}
{{64,22,5085696},{88,84,5085696}}
{{64,24,5406720},{88,8,5406720}}
{{74,14,5470080},{74,66,5470080},{80,74,5470080}}
{{63,27,5511240},{63,45,5511240},{72,63,5511240}}
{{65,23,5525520},{80,11,5525520}}
{{78,12,5559840},{99,96,5559840}}
{{62,34,5666304},{68,56,5666304}}
{{70,18,5765760},{77,13,5765760}}
{{63,30,5800410},{99,6,5800410}}
{{64,46,5829120},{92,88,5829120}}
{{63,32,5937120},{76,14,5937120}}
{{63,33,5987520},{66,24,5987520}}
{{65,26,5997810},{78,13,5997810}}
{{73,17,6254640},{73,63,6254640},{80,73,6254640}}
{{70,21,6554730},{98,7,6554730}}
{{69,55,6588120},{77,15,6588120}}
{{65,33,6726720},{88,10,6726720}}
{{66,46,6800640},{70,22,6800640}}
{{65,35,6825000},{65,40,6825000},{75,65,6825000}}
{{66,30,6842880},{72,60,6842880}}
{{72,21,7171416},{93,9,7171416}}
{{67,32,7428960},{67,45,7428960},{77,67,7428960}}
{{90,84,7892640},{96,9,7892640}}
{{70,28,8067360},{84,14,8067360}}
{{68,42,8168160},{91,11,8168160},{91,85,8168160},{96,91,8168160}}
{{76,20,8171520},{76,64,8171520},{84,76,8171520}}
{{70,30,8400000},{70,50,8400000},{80,70,8400000}}
{{76,21,8514660},{77,20,8514660}}
{{69,36,8607060},{84,15,8607060}}
{{72,27,8660520},{99,9,8660520}}
{{77,64,9033024},{94,88,9033024}}
{{75,30,10631250},{90,15,10631250}}
{{80,24,11182080},{84,20,11182080}}
{{75,35,11550000},{80,25,11550000}}
{{77,33,12298440},{77,55,12298440},{88,77,12298440}}
{{77,35,12677280},{84,70,12677280}}
{{77,51,13069056},{78,34,13069056}}
{{77,38,13123110},{78,55,13123110}}
{{91,19,13693680},{91,80,13693680},{99,91,13693680}}
{{80,32,13762560},{96,16,13762560}}
{{80,58,14087040},{87,23,14087040}}
{{78,42,14152320},{78,48,14152320},{90,78,14152320}}
{{79,40,14665560},{79,51,14665560},{91,79,14665560}}
{{86,26,15025920},{86,70,15025920},{96,86,15025920}}
{{92,22,16151520},{96,19,16151520}}
{{84,36,17418240},{84,60,17418240},{96,84,17418240}}
{{84,44,18923520},{88,32,18923520}}
{{85,57,19263720},{95,24,19263720}}
{{95,25,19950000},{95,80,19950000}}
{{87,56,21597576},{91,33,21597576}}
{{88,40,21626880},{96,80,21626880}}
{{88,42,22102080},{91,69,22102080}}
{{88,57,22546920},{95,29,22546920}}
{{88,45,22647240},{95,77,22647240}}
{{88,52,23063040},{90,64,23063040}}
{{93,33,23201640},{93,72,23201640}}
{{90,62,23748480},{95,31,23748480}}
{{90,42,23950080},{96,30,23950080}}
{{91,39,23991240},{91,65,23991240}}
{{91,49,26218920},{91,56,26218920}}
{{98,32,26906880},{98,78,26906880}}
{{98,42,32269440},{98,70,32269440}}
{{97,55,34058640},{97,57,34058640}}

Paul.

2. paul says:

Here are the solutions where there would be 10 pairs of (a, b, x, y) and b < a <= 1000

{403,115,6913932480},
{403,333,6913932480},
{414,104,6913932480},
{448,403,6913932480},
{558,40,6913932480}.

and

{518,288,27655729920},
{518,310,27655729920},
{598,518,27655729920},
{736,70,27655729920},
{851,45,27655729920}.

Paul.

3. pipo says:

There are a lot:
The smallest with 3 solutions (1, 15), (2, 12) and (4,10) = 3360
Smallest with 4 soltions: (33, 37) , (37, 40), (55, 56), (7,37) = 341880

210 1 6 2 5
840 3 7 5 7
840 5 7 7 8
1320 1 11 3 8
2730 1 14 3 10
3360 1 15 2 12
3360 2 12 4 10
5280 10 12 5 11
7980 1 20 7 12
10920 13 15 7 13
10920 7 13 8 13
13440 10 14 14 16
13440 14 16 6 14
17010 3 18 6 15
18480 5 16 7 15
21120 2 22 6 16
31920 16 19 19 21
31920 19 21 5 19
43680 2 28 5 21
43680 5 21 6 20
53760 2 30 4 24
53760 4 24 8 20
68040 15 21 21 24
68040 21 24 9 21
68640 32 33 4 26
71610 10 21 2 33
73920 11 21 8 22
84480 10 22 20 24
85470 15 22 2 35
88536 3 31 7 24
97440 28 30 3 32
106260 12 23 5 28
106920 3 33 9 24
114114 38 39 7 26
127680 14 24 2 40
127680 2 40 3 35
131250 10 25 5 30
174720 14 26 16 26
174720 16 26 26 30
175560 1 56 5 33
215040 12 28 20 28
215040 20 28 28 32
221130 3 42 9 30
234780 15 28 4 39
272160 12 30 3 45
272160 3 45 6 36
274560 1 65 22 30
286440 11 31 24 31
286440 24 31 31 35
295680 10 32 14 30
317856 11 32 42 44
337920 12 32 4 44
341880 33 37 37 40
341880 37 40 55 56
341880 55 56 7 37
354144 17 31 28 34
364320 23 32 44 46
389760 2 58 29 35
420420 3 52 5 44
425040 11 35 23 33
427680 15 33 30 36
456456 1 77 19 33
499590 2 63 26 35
504210 14 35 7 42
510720 10 38 32 38
510720 32 38 38 42
525000 15 35 25 35
525000 25 35 35 40
614040 1 85 8 43
646380 21 36 3 60
698880 10 42 12 40
698880 12 40 4 56
702240 28 38 55 57
825000 15 40 5 55
860160 16 40 4 60
860160 4 60 8 48
884520 21 39 24 39
884520 24 39 39 45
939120 13 43 35 43
939120 35 43 43 48
1088640 18 42 30 42
1088640 30 42 42 48
1098240 64 66 8 52
1145760 20 42 4 66
1145760 4 66 7 55
1182720 16 44 22 42
1271424 2 86 21 43
1351680 20 44 40 48
1367520 3 77 30 44
1367520 30 44 4 70
1377810 18 45 9 54
1416576 14 48 6 62
1441440 26 44 32 45
1457280 2 90 9 55
1496880 15 48 21 45
1559040 56 60 6 64
1563660 28 45 5 68
1647030 31 46 7 62
1681680 16 49 39 49
1681680 39 49 49 55
1700160 10 56 24 46
1706250 15 50 5 70
1710720 18 48 6 66
1825824 14 52 76 78
1857240 11 56 5 72
1998360 56 61 61 65
1998360 61 65 9 61
2016840 21 49 35 49
2016840 35 49 49 56
2042040 13 55 3 88
2042040 3 88 40 51
2042880 28 48 4 80
2042880 4 80 6 70
2100000 10 60 20 50
2100000 20 50 5 75
2456160 35 51 84 86
2585520 15 57 48 57
2585520 48 57 57 63
2795520 28 52 32 52
2795520 32 52 52 60
3074610 11 66 22 55
3169320 21 56 7 77
3267264 13 64 22 56
3300000 25 55 50 60
3423420 36 55 4 95
3440640 24 56 40 56
3440640 40 56 56 64
3521760 11 69 32 55
3538080 15 63 18 60
3538080 18 60 6 84
3666390 14 65 6 85
3756480 30 56 5 91
3756480 5 91 8 78
3769920 34 56 55 64
4354560 12 72 24 60
4354560 24 60 6 90
4583040 22 62 48 62
4583040 48 62 62 70
4730880 20 64 28 60
5085696 22 64 84 88
5406720 24 64 8 88
5470080 14 74 66 74
5470080 66 74 74 80
5511240 27 63 45 63
5511240 45 63 63 72
5525520 11 80 23 65
5559840 12 78 96 99
5666304 34 62 56 68
5765760 13 77 18 70
5800410 30 63 6 99
5829120 46 64 88 92
5937120 14 76 32 63
5987520 24 66 33 63
5997810 13 78 26 65
6254640 17 73 63 73
6254640 63 73 73 80
6554730 21 70 7 98
6588120 15 77 55 69
6726720 10 88 33 65
6800640 22 70 46 66
6825000 35 65 40 65
6825000 40 65 65 75
6842880 30 66 60 72
7171416 21 72 9 93
7428960 32 67 45 67
7428960 45 67 67 77
7892640 84 90 9 96
8067360 14 84 28 70
8168160 11 91 42 68
8168160 42 68 85 91
8168160 85 91 91 96
8171520 20 76 64 76
8171520 64 76 76 84
8400000 30 70 50 70
8400000 50 70 70 80
8514660 20 77 21 76
8607060 15 84 36 69
8660520 27 72 9 99
9033024 64 77 88 94
10631250 15 90 30 75
11182080 20 84 24 80
11550000 25 80 35 75
12298440 33 77 55 77
12298440 55 77 77 88
12677280 35 77 70 84
13069056 34 78 51 77
13123110 38 77 55 78
13693680 19 91 80 91
13693680 80 91 91 99
13762560 16 96 32 80
14087040 23 87 58 80
14152320 42 78 48 78
14152320 48 78 78 90
14665560 40 79 51 79
14665560 51 79 79 91
15025920 26 86 70 86
15025920 70 86 86 96
16151520 19 96 22 92
17418240 36 84 60 84
17418240 60 84 84 96
18923520 32 88 44 84
19263720 24 95 57 85
19950000 25 95 80 95
21597576 33 91 56 87
21626880 40 88 80 96
22102080 42 88 69 91
22546920 29 95 57 88
22647240 45 88 77 95
23063040 52 88 64 90
23201640 33 93 72 93
23748480 31 95 62 90
23950080 30 96 42 90
23991240 39 91 65 91
26218920 49 91 56 91
26906880 32 98 78 98
32269440 42 98 70 98
34058640 55 97 57 97

pipo

4. pipo says:

Oops, just too late

pipo

• paul says:

840 has 3 solutions too, smaller than 3360.
P.

• benvitalis says:

Thanks Pipo, Paul. This gives us the product of both legs in Pythagorean triangles.
Now, can you find 2 Pythagorean triples such that
x^2 + y^2 + z^2 = a^2 + b^2 + c^2, and
x*y*z = a*b*c