## a^4 + 14(a*b)^2 + b^4 = c^4 + 14(c*d)^2 + d^4

I like this type of equations,
$a^4 \; + \; 2 \, n \, (a \, b)^2 \; + \; b^4 \; = \; c^4 \; + \; 2 \, n \, (c \, d)^2 \; + \; d^4$

because they can be used to form identities.

Let’s solve for   n = 7

$a^4 \; + \; 14 \: a^2 \: b^2 \; + \; b^4 \; = \; c^4 \; + \; 14 \: c^2 \: d^2 \; + \; d^4$

where   $a, b, c, d$   are distinct positive integers

For example,

$63^4 + 14(63\times 235)^2 + 235^4 \; = \; 107^4 + 14(107\times 177)^2 + 177^4$
$63^4 + 14\times 14805^2 + 235^4 \; = \; 107^4 + 14\times 18939^2 + 177^4$

Note that

$(a + b)^8 \; + \; (a - b)^8 \; + \; (4 \, a \, b)^4$
$= \; 2 \, a^8 \; + \; 56 \, a^6 \, b^2 \; + \; 396 \, a^4 \, b^4 \; + \; 56 \, a^2 \, b^6 \; + \; 2 \, b^8$
$= \; 2 \, (a^4 \; + \; 14 \, a^2 \, b^2 \; + \; b^4)^2$

So,

$(a+b)^8 \; + \; (a-b)^8 \; + \; (4 \, a \, b)^4 \; = \; (c+d)^8 \; + \; (c-d)^8 \; + \; (4 \, c \, d)^4$
$2 \, (a^4 \; + \; 14 \, a^2 \, b^2 \; + \; b^4)^2 \; = \; 2 \, (c^4 \; + \; 14 \, c^2 \, d^2 \; + \; d^4)^2$
$(a^4 \; + \; 14 \, a^2 \, b^2 \; + \; b^4)^2 \; = \; (c^4 \; + \; 14 \, c^2 \, d^2 \; + \; d^4)^2$

For example,

(1)
$1^4 \; + \; 14(1\times 9)^2 \; + \; 9^4 \; = \; 2^4 \; + \; 14(2\times 8)^2 \; + \; 8^4$
becomes
$10^8 + (4\times 9)^4 + 8^8 \; = \; 10^8 + (4\times 16)^4 + 6^8$
$10^8 + 6^8 + 8^8 \; = \; 10^8 + 8^8 + 6^8$

(2)
$1^4 + 14(1\times 25)^2 + 25^4 \; = \; 8^4 + 14(8\times 18)^2 + 18^4$

$26^8 + (4\times 25)^4 + 24^8 \; = \; 26^8 + (4\times 8\times 18)^4 + 10^8$
$26^8 + 100^4 + 24^8 \; = \; 26^8 + 576^4 + 10^8$
$26^8 + 10^8 + 24^8 \; = \; 26^8 + 24^8 + 10^8$

(3)
$63^4 + 14(63\times 235)^2 + 235^4 \; = \; 107^4 + 14(107\times 177)^2 + 177^4$

$(63+235)^8 + (4\times 63\times 235)^4 + (63-235)^8$
$= \;(107+177)^8 + (4\times 107\times 177)^4 + (107-177)^8$

$298^8 + 59220^4 + 172^8 \; = \; 284^8 + 75756^4 + 70^8$

(4)
$65^4 + 14(65\times 215)^2 + 215^4 \; = \; 95^4 + 14(95\times 175)^2 + 175^4$

$(65+215)^8 + (4\times 65\times 215)^4 + (65-215)^8$
$= \; (95+175)^8 + (4\times 95\times 175)^4 + (95-175)^8$

$280^8 + 55900^4 + 150^8 \; = \; 270^8 + 66500^4 + 80^8$

(5)
$76^4 + 14(76\times 240)^2 + 240^4 \; = \; 134^4 + 14(134\times 166)^2 + 166^4$

$(76+240)^8 + (4\times 76\times 240)^4 + (76-240)^8$
$= \; (134+166)^8 + (4\times 134\times 166)^4 + (134-166)^8$

$316^8 + 72960^4 + 164^8 \; = \; 300^8 + 88976^4 + 32^8$

(6)
$79^4 + 14(79\times 151)^2 + 151^4 \; = \; 96^4 + 14(96\times 130)^2 + 130^4$

$(79+151)^8 + (4\times 79\times 151)^4 + (79-151)^8$
$= \; (96+130)^8 + (4\times 96\times 130)^4 + (96-130)^8$

$230^8 + 47716^4 + 72^8 \; = \; 226^8 + 49920^4 + 34^8$

math grad - Interest: Number theory
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### 2 Responses to a^4 + 14(a*b)^2 + b^4 = c^4 + 14(c*d)^2 + d^4

1. paul says:

Here are all the solutions with “a” less than the example and 3 more for good measure

1^4 + 14(1 x 9)^2 + 9^4 = 2^4 + 14(2 x 8)^2 + 8^4 = 7696
1^4 + 14(1 x 25)^2 + 25^4 = 8^4 + 14(8 x 18)^2 + 18^4 = 399376
1^4 + 14(1 x 49)^2 + 49^4 = 18^4 + 14(18 x 32)^2 + 32^4 = 5798416
1^4 + 14(1 x 81)^2 + 81^4 = 32^4 + 14(32 x 50)^2 + 50^4 = 43138576
1^4 + 14(1 x 121)^2 + 121^4 = 50^4 + 14(50 x 72)^2 + 72^4 = 214563856
1^4 + 14(1 x 169)^2 + 169^4 = 72^4 + 14(72 x 98)^2 + 98^4 = 816130576
1^4 + 14(1 x 225)^2 + 225^4 = 98^4 + 14(98 x 128)^2 + 128^4 = 2563599376
2^4 + 14(2 x 18)^2 + 18^4 = 4^4 + 14(4 x 16)^2 + 16^4 = 123136
2^4 + 14(2 x 32)^2 + 32^4 = 9^4 + 14(9 x 25)^2 + 25^4 = 1105936
2^4 + 14(2 x 50)^2 + 50^4 = 16^4 + 14(16 x 36)^2 + 36^4 = 6390016
2^4 + 14(2 x 72)^2 + 72^4 = 25^4 + 14(25 x 49)^2 + 49^4 = 27164176
2^4 + 14(2 x 96)^2 + 96^4 = 23^4 + 14(23 x 79)^2 + 79^4 = 85450768
2^4 + 14(2 x 98)^2 + 98^4 = 36^4 + 14(36 x 64)^2 + 64^4 = 92774656
2^4 + 14(2 x 128)^2 + 128^4 = 49^4 + 14(49 x 81)^2 + 81^4 = 269352976
2^4 + 14(2 x 162)^2 + 162^4 = 64^4 + 14(64 x 100)^2 + 100^4 = 690217216
2^4 + 14(2 x 200)^2 + 200^4 = 81^4 + 14(81 x 121)^2 + 121^4 = 1602240016
2^4 + 14(2 x 242)^2 + 242^4 = 100^4 + 14(100 x 144)^2 + 144^4 = 3433021696
3^4 + 14(3 x 27)^2 + 27^4 = 6^4 + 14(6 x 24)^2 + 24^4 = 623376
3^4 + 14(3 x 75)^2 + 75^4 = 24^4 + 14(24 x 54)^2 + 54^4 = 32349456
3^4 + 14(3 x 147)^2 + 147^4 = 54^4 + 14(54 x 96)^2 + 96^4 = 469671696
3^4 + 14(3 x 243)^2 + 243^4 = 96^4 + 14(96 x 150)^2 + 150^4 = 3494224656
4^4 + 14(4 x 36)^2 + 36^4 = 8^4 + 14(8 x 32)^2 + 32^4 = 1970176
4^4 + 14(4 x 64)^2 + 64^4 = 18^4 + 14(18 x 50)^2 + 50^4 = 17694976
4^4 + 14(4 x 100)^2 + 100^4 = 32^4 + 14(32 x 72)^2 + 72^4 = 102240256
4^4 + 14(4 x 144)^2 + 144^4 = 50^4 + 14(50 x 98)^2 + 98^4 = 434626816
4^4 + 14(4 x 192)^2 + 192^4 = 46^4 + 14(46 x 158)^2 + 158^4 = 1367212288
4^4 + 14(4 x 196)^2 + 196^4 = 72^4 + 14(72 x 128)^2 + 128^4 = 1484394496
5^4 + 14(5 x 45)^2 + 45^4 = 10^4 + 14(10 x 40)^2 + 40^4 = 4810000
5^4 + 14(5 x 125)^2 + 125^4 = 40^4 + 14(40 x 90)^2 + 90^4 = 249610000
5^4 + 14(5 x 245)^2 + 245^4 = 90^4 + 14(90 x 160)^2 + 160^4 = 3624010000
6^4 + 14(6 x 54)^2 + 54^4 = 12^4 + 14(12 x 48)^2 + 48^4 = 9974016
6^4 + 14(6 x 96)^2 + 96^4 = 27^4 + 14(27 x 75)^2 + 75^4 = 89580816
6^4 + 14(6 x 150)^2 + 150^4 = 48^4 + 14(48 x 108)^2 + 108^4 = 517591296
6^4 + 14(6 x 152)^2 + 152^4 = 13^4 + 14(13 x 149)^2 + 149^4 = 545440528
6^4 + 14(6 x 216)^2 + 216^4 = 75^4 + 14(75 x 147)^2 + 147^4 = 2200298256
7^4 + 14(7 x 63)^2 + 63^4 = 14^4 + 14(14 x 56)^2 + 56^4 = 18478096
7^4 + 14(7 x 175)^2 + 175^4 = 56^4 + 14(56 x 126)^2 + 126^4 = 958901776
8^4 + 14(8 x 50)^2 + 50^4 = 9^4 + 14(9 x 49)^2 + 49^4 = 8494096
8^4 + 14(8 x 72)^2 + 72^4 = 16^4 + 14(16 x 64)^2 + 64^4 = 31522816
8^4 + 14(8 x 98)^2 + 98^4 = 25^4 + 14(25 x 81)^2 + 81^4 = 100846096
8^4 + 14(8 x 128)^2 + 128^4 = 36^4 + 14(36 x 100)^2 + 100^4 = 283119616
8^4 + 14(8 x 150)^2 + 150^4 = 67^4 + 14(67 x 85)^2 + 85^4 = 526414096
8^4 + 14(8 x 162)^2 + 162^4 = 49^4 + 14(49 x 121)^2 + 121^4 = 712266256
8^4 + 14(8 x 200)^2 + 200^4 = 64^4 + 14(64 x 144)^2 + 144^4 = 1635844096
8^4 + 14(8 x 242)^2 + 242^4 = 81^4 + 14(81 x 169)^2 + 169^4 = 3482219536
9^4 + 14(9 x 81)^2 + 81^4 = 18^4 + 14(18 x 72)^2 + 72^4 = 50493456
9^4 + 14(9 x 121)^2 + 121^4 = 32^4 + 14(32 x 98)^2 + 98^4 = 230968336
9^4 + 14(9 x 169)^2 + 169^4 = 50^4 + 14(50 x 128)^2 + 128^4 = 848125456
9^4 + 14(9 x 225)^2 + 225^4 = 72^4 + 14(72 x 162)^2 + 162^4 = 2620305936
10^4 + 14(10 x 90)^2 + 90^4 = 20^4 + 14(20 x 80)^2 + 80^4 = 76960000
10^4 + 14(10 x 160)^2 + 160^4 = 45^4 + 14(45 x 125)^2 + 125^4 = 691210000
10^4 + 14(10 x 250)^2 + 250^4 = 80^4 + 14(80 x 180)^2 + 180^4 = 3993760000
11^4 + 14(11 x 99)^2 + 99^4 = 22^4 + 14(22 x 88)^2 + 88^4 = 112677136
12^4 + 14(12 x 108)^2 + 108^4 = 24^4 + 14(24 x 96)^2 + 96^4 = 159584256
12^4 + 14(12 x 192)^2 + 192^4 = 54^4 + 14(54 x 150)^2 + 150^4 = 1433293056
13^4 + 14(13 x 43)^2 + 43^4 = 19^4 + 14(19 x 35)^2 + 35^4 = 7822096
13^4 + 14(13 x 117)^2 + 117^4 = 26^4 + 14(26 x 104)^2 + 104^4 = 219805456
14^4 + 14(14 x 126)^2 + 126^4 = 28^4 + 14(28 x 112)^2 + 112^4 = 295649536
14^4 + 14(14 x 224)^2 + 224^4 = 63^4 + 14(63 x 175)^2 + 175^4 = 2655352336
15^4 + 14(15 x 135)^2 + 135^4 = 30^4 + 14(30 x 120)^2 + 120^4 = 389610000
16^4 + 14(16 x 100)^2 + 100^4 = 18^4 + 14(18 x 98)^2 + 98^4 = 135905536
16^4 + 14(16 x 144)^2 + 144^4 = 32^4 + 14(32 x 128)^2 + 128^4 = 504365056
16^4 + 14(16 x 196)^2 + 196^4 = 50^4 + 14(50 x 162)^2 + 162^4 = 1613537536
17^4 + 14(17 x 153)^2 + 153^4 = 34^4 + 14(34 x 136)^2 + 136^4 = 642777616
18^4 + 14(18 x 128)^2 + 128^4 = 25^4 + 14(25 x 121)^2 + 121^4 = 342858256
18^4 + 14(18 x 162)^2 + 162^4 = 36^4 + 14(36 x 144)^2 + 144^4 = 807895296
18^4 + 14(18 x 200)^2 + 200^4 = 49^4 + 14(49 x 169)^2 + 169^4 = 1781544976
18^4 + 14(18 x 242)^2 + 242^4 = 64^4 + 14(64 x 196)^2 + 196^4 = 3695493376
19^4 + 14(19 x 171)^2 + 171^4 = 38^4 + 14(38 x 152)^2 + 152^4 = 1002950416
20^4 + 14(20 x 180)^2 + 180^4 = 40^4 + 14(40 x 160)^2 + 160^4 = 1231360000
21^4 + 14(21 x 189)^2 + 189^4 = 42^4 + 14(42 x 168)^2 + 168^4 = 1496725776
22^4 + 14(22 x 198)^2 + 198^4 = 44^4 + 14(44 x 176)^2 + 176^4 = 1802834176
23^4 + 14(23 x 207)^2 + 207^4 = 46^4 + 14(46 x 184)^2 + 184^4 = 2153656336
24^4 + 14(24 x 150)^2 + 150^4 = 27^4 + 14(27 x 147)^2 + 147^4 = 688021776
24^4 + 14(24 x 216)^2 + 216^4 = 48^4 + 14(48 x 192)^2 + 192^4 = 2553348096
25^4 + 14(25 x 169)^2 + 169^4 = 32^4 + 14(32 x 162)^2 + 162^4 = 1066030096
25^4 + 14(25 x 225)^2 + 225^4 = 50^4 + 14(50 x 200)^2 + 200^4 = 3006250000
26^4 + 14(26 x 86)^2 + 86^4 = 38^4 + 14(38 x 70)^2 + 70^4 = 125153536
26^4 + 14(26 x 234)^2 + 234^4 = 52^4 + 14(52 x 208)^2 + 208^4 = 3516887296
27^4 + 14(27 x 243)^2 + 243^4 = 54^4 + 14(54 x 216)^2 + 216^4 = 4089969936
32^4 + 14(32 x 200)^2 + 200^4 = 36^4 + 14(36 x 196)^2 + 196^4 = 2174488576
32^4 + 14(32 x 242)^2 + 242^4 = 49^4 + 14(49 x 225)^2 + 225^4 = 4270364176
38^4 + 14(38 x 120)^2 + 120^4 = 67^4 + 14(67 x 83)^2 + 83^4 = 500555536
39^4 + 14(39 x 129)^2 + 129^4 = 57^4 + 14(57 x 105)^2 + 105^4 = 633589776
40^4 + 14(40 x 250)^2 + 250^4 = 45^4 + 14(45 x 245)^2 + 245^4 = 5308810000
52^4 + 14(52 x 172)^2 + 172^4 = 76^4 + 14(76 x 140)^2 + 140^4 = 2002456576
63^4 + 14(63 x 235)^2 + 235^4 = 107^4 + 14(107 x 177)^2 + 177^4 = 6134185936
65^4 + 14(65 x 215)^2 + 215^4 = 95^4 + 14(95 x 175)^2 + 175^4 = 4888810000
76^4 + 14(76 x 240)^2 + 240^4 = 134^4 + 14(134 x 166)^2 + 166^4 = 8008888576
79^4 + 14(79 x 151)^2 + 151^4 = 96^4 + 14(96 x 130)^2 + 130^4 = 2551050256

Paul.