## Equal square sums and products

Equal square sums and products

$x^2 \; + \; y^2 \; + \; z^2 \; = \; a^2 \; + \; b^2 \; + \; c^2$

$x \cdot y \cdot z \; = \; a \cdot b \cdot c$

Here’s one possible solution:

$2^2 \; + \; 13^2 \; + \; 24^2 \; = \; 3^2 \; + \; 8^2 \; + \; 26^2 \; = \; 749$

$2 \times 13 \times 24 \; = \; 3 \times 8 \times 26 \; = \; 624$

Find other solutions.

Related topic:   Sums & Products| x+y+z=a+b+c and x*y*z=a*b*c

……………………….. Part 1 ………………………..

(1)   parametric solution of :

$x^2 \; + \; y^2 \; + \; z^2 \; = \; a^2 \; + \; b^2 \; + \; c^2$

$x \cdot y \cdot z \; = \; a \cdot b \cdot c$

$\begin{pmatrix} \:a & \: b & \: c \: \\ \:d & \: e & \: f \: \\ \:g & \: h & \: k \: \end{pmatrix}$

We try
row products ………   $x = a b c$,   $y = d e f$,   $z = g h k$
column products …   $u = a d g$,   $v = b e h$,   $w = c f k$

This will work if we choose   $a, b, c, d, e, f, g, h, k$   to satisfy the side condition:

$a^2 \, (b^2 \, c^2 - d^2 \, g^2) \; + \; e^2 \, (d^2 \, f^2 - b^2 \, h^2) \; + \; k^2 \, (g^2 \, h^2 - c^2 \, f^2) \; = \; 0$

Suitable matices are

$\begin{pmatrix} \;1 \; & \; 2q - p \; & \; p - q \; \\ \;p \; & \; 1 \; & \; 2p - q \; \\ \;p+q \; & \; q \; & \; 1 \; \end{pmatrix}$

$\begin{pmatrix} \;1 \; & \; 1 \; & \; 2p + 1 \; \\ \;1 \; & \; 1 \; & \; 3p - 2 \; \\ \;p \; & \; 6p - 1 \; & \; 1 \end{pmatrix}$

and

$\begin{pmatrix} \;1 \; & \; 1 \; & \; 2p - 1 \; \\ \;1 \; & \; 1 \; & \; 3p + 2 \; \\ \;p \; & \; 6p+1 \; & \; 1 \end{pmatrix}$

where   $p$   and   $q$   are integers.

(2)   parametric solution:

$x \; = \; v \,(w^2 \; - \; v^2 \; + \; u \, w)$
$y \; = \; (u \; + \; w) \,(v^2 \; - \; w^2 \; + \; u \, w)$
$z \; = \; u \, v \, (u \; - \; w)$

$a \; = \; v \,(v^2 \; - \; w^2 \; + \; u \, w)$
$b \; = \; (u \; - \; w) \,(w^2 \; - \; v^2 \; + \; u \, w)$
$c \; = \; u \, v \, (u \; + \; w)$

……………………….. Part 2 ………………………..

If

$a^2 \; + \; b^2 \; + \; c^2 \; = \; d^2 \; + \; e^2 \; + \; f^2$
$a \cdot b \cdot c \; = \; d \cdot e \cdot f$

then,

$(a+b+c) \; + \; (a-b-c) \; + \; (-a-b+c) \; + \; (-a+b-c)$
$= \; (d+e+f) \; + \; (d-e-f) \; + \; (-d-e+f) \; + \; (-d+e-f) \; = \; 0$

$(a+b+c)^2 \; + \; (a-b-c)^2 \; + \; (-a-b+c)^2 \; + \; (-a+b-c)^2$
$= \; (d+e+f)^2 \; + \; (d-e-f)^2 + (-d-e+f)^2 \; + \; (-d+e-f)^2$

$(a+b+c)^3 \; + \; (a-b-c)^3 \; + \; (-a-b+c)^3 \; + \; (-a+b-c)^3$
$= \; (d+e+f)^3 \; + \; (d-e-f)^3 \; + \; (-d-e+f)^3 \; + \; (-d+e-f)^3$

$(a+b+c)^5 \; + \; (a-b-c)^5 \; + \; (-a-b+c)^5 \; + \; (-a+b-c)^5$
$= \; (d+e+f)^5 \; + \; (d-e-f)^5 \; + \; (-d-e+f)^5 \; + \; (-d+e-f)^5$

$1^2 + 10^2 + 12^2 \; = \; 2^2 + 4^2 + 15^2 \; = \; 245$
$1 \times 10 \times 12 \; = \; 2 \times 4 \times 15 \; = \; 120$

a = 1,   b = 10,   c = 12   ……..   d = 2,   e = 4,   f = 15
a+b+c = 23   ……..   d+e+f = 21
a-b-c = -21   ………   d-e-f = -17
-a-b+c = 1   ……….   -d-e+f = 9
-a+b-c = -3   ……..   -d+e-f = -13

$23 + (-21) + 1 + (-3) \; = \; 21 + (-17) + 9 + (-13) = 0$
$23^2 + (-21)^2 + 1^2 + (-3)^2 \; = \; 21^2 + (-17)^2 + 9^2 + (-13)^2 = 980$
$23^3 + (-21)^3 + 1^3 + (-3)^3 \; = \; 21^3 + (-17)^3 + 9^3 + (-13)^3 = 2880$
$23^5 + (-21)^5 + 1^5 + (-3)^5 \; = \; 21^5 + (-17)^5 + 9^5 + (-13)^5 = 2352000$

math grad - Interest: Number theory
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### 2 Responses to Equal square sums and products

1. paul says:

Here is a list with a b c <=10000

1^2 + 10^2 + 12^2 = 2^2 + 4^2 + 15^2 = 245
1 x 10 x 12 = 2 x 4 x 15 = 120

2^2 + 11^2 + 20^2 = 4^2 + 5^2 + 22^2 = 525
2 x 11 x 20 = 4 x 5 x 22 = 440

1^2 + 17^2 + 30^2 = 3^2 + 5^2 + 34^2 = 1190
1 x 17 x 30 = 3 x 5 x 34 = 510

2^2 + 13^2 + 24^2 = 3^2 + 8^2 + 26^2 = 749
2 x 13 x 24 = 3 x 8 x 26 = 624

2^2 + 20^2 + 21^2 = 5^2 + 6^2 + 28^2 = 845
2 x 20 x 21 = 5 x 6 x 28 = 840

2^2 + 20^2 + 24^2 = 4^2 + 8^2 + 30^2 = 980
2 x 20 x 24 = 4 x 8 x 30 = 960
3^2 + 16^2 + 20^2 = 5^2 + 8^2 + 24^2 = 665
3 x 16 x 20 = 5 x 8 x 24 = 960

1^2 + 28^2 + 42^2 = 2^2 + 12^2 + 49^2 = 2549
1 x 28 x 42 = 2 x 12 x 49 = 1176

1^2 + 33^2 + 45^2 = 3^2 + 9^2 + 55^2 = 3115
1 x 33 x 45 = 3 x 9 x 55 = 1485

1^2 + 23^2 + 66^2 = 2^2 + 11^2 + 69^2 = 4886
1 x 23 x 66 = 2 x 11 x 69 = 1518

1^2 + 26^2 + 60^2 = 4^2 + 6^2 + 65^2 = 4277
1 x 26 x 60 = 4 x 6 x 65 = 1560
3^2 + 20^2 + 26^2 = 4^2 + 13^2 + 30^2 = 1085
3 x 20 x 26 = 4 x 13 x 30 = 1560

1^2 + 40^2 + 42^2 = 2^2 + 15^2 + 56^2 = 3365
1 x 40 x 42 = 2 x 15 x 56 = 1680
2^2 + 24^2 + 35^2 = 3^2 + 14^2 + 40^2 = 1805
2 x 24 x 35 = 3 x 14 x 40 = 1680

1^2 + 32^2 + 56^2 = 4^2 + 7^2 + 64^2 = 4161
1 x 32 x 56 = 4 x 7 x 64 = 1792

4^2 + 21^2 + 22^2 = 6^2 + 11^2 + 28^2 = 941
4 x 21 x 22 = 6 x 11 x 28 = 1848

1^2 + 40^2 + 52^2 = 4^2 + 8^2 + 65^2 = 4305
1 x 40 x 52 = 4 x 8 x 65 = 2080

1^2 + 33^2 + 70^2 = 5^2 + 6^2 + 77^2 = 5990
1 x 33 x 70 = 5 x 6 x 77 = 2310

1^2 + 38^2 + 68^2 = 2^2 + 17^2 + 76^2 = 6069
1 x 38 x 68 = 2 x 17 x 76 = 2584

5^2 + 22^2 + 24^2 = 8^2 + 11^2 + 30^2 = 1085
5 x 22 x 24 = 8 x 11 x 30 = 2640

2^2 + 36^2 + 39^2 = 6^2 + 9^2 + 52^2 = 2821
2 x 36 x 39 = 6 x 9 x 52 = 2808

1^2 + 52^2 + 57^2 = 3^2 + 13^2 + 76^2 = 5954
1 x 52 x 57 = 3 x 13 x 76 = 2964

2^2 + 29^2 + 52^2 = 4^2 + 13^2 + 58^2 = 3549
2 x 29 x 52 = 4 x 13 x 58 = 3016

3^2 + 19^2 + 55^2 = 5^2 + 11^2 + 57^2 = 3395
3 x 19 x 55 = 5 x 11 x 57 = 3135

2^2 + 36^2 + 45^2 = 3^2 + 20^2 + 54^2 = 3325
2 x 36 x 45 = 3 x 20 x 54 = 3240
3^2 + 30^2 + 36^2 = 6^2 + 12^2 + 45^2 = 2205
3 x 30 x 36 = 6 x 12 x 45 = 3240
4^2 + 27^2 + 30^2 = 5^2 + 18^2 + 36^2 = 1645
4 x 27 x 30 = 5 x 18 x 36 = 3240

2^2 + 31^2 + 55^2 = 5^2 + 11^2 + 62^2 = 3990
2 x 31 x 55 = 5 x 11 x 62 = 3410

4^2 + 22^2 + 40^2 = 8^2 + 10^2 + 44^2 = 2100
4 x 22 x 40 = 8 x 10 x 44 = 3520

2^2 + 31^2 + 57^2 = 3^2 + 19^2 + 62^2 = 4214
2 x 31 x 57 = 3 x 19 x 62 = 3534

1^2 + 49^2 + 78^2 = 3^2 + 14^2 + 91^2 = 8486
1 x 49 x 78 = 3 x 14 x 91 = 3822

4^2 + 23^2 + 42^2 = 7^2 + 12^2 + 46^2 = 2309
4 x 23 x 42 = 7 x 12 x 46 = 3864

1^2 + 37^2 + 105^2 = 5^2 + 7^2 + 111^2 = 12395
1 x 37 x 105 = 5 x 7 x 111 = 3885

2^2 + 34^2 + 60^2 = 6^2 + 10^2 + 68^2 = 4760
2 x 34 x 60 = 6 x 10 x 68 = 4080
3^2 + 34^2 + 40^2 = 8^2 + 10^2 + 51^2 = 2765
3 x 34 x 40 = 8 x 10 x 51 = 4080

2^2 + 39^2 + 55^2 = 5^2 + 13^2 + 66^2 = 4550
2 x 39 x 55 = 5 x 13 x 66 = 4290

3^2 + 34^2 + 44^2 = 4^2 + 22^2 + 51^2 = 3101
3 x 34 x 44 = 4 x 22 x 51 = 4488
4^2 + 33^2 + 34^2 = 6^2 + 17^2 + 44^2 = 2261
4 x 33 x 34 = 6 x 17 x 44 = 4488

4^2 + 26^2 + 48^2 = 6^2 + 16^2 + 52^2 = 2996
4 x 26 x 48 = 6 x 16 x 52 = 4992

5^2 + 27^2 + 39^2 = 9^2 + 13^2 + 45^2 = 2275
5 x 27 x 39 = 9 x 13 x 45 = 5265

2^2 + 51^2 + 52^2 = 3^2 + 26^2 + 68^2 = 5309
2 x 51 x 52 = 3 x 26 x 68 = 5304

3^2 + 35^2 + 54^2 = 9^2 + 10^2 + 63^2 = 4150
3 x 35 x 54 = 9 x 10 x 63 = 5670

5^2 + 32^2 + 36^2 = 8^2 + 16^2 + 45^2 = 2345
5 x 32 x 36 = 8 x 16 x 45 = 5760

2^2 + 41^2 + 72^2 = 8^2 + 9^2 + 82^2 = 6869
2 x 41 x 72 = 8 x 9 x 82 = 5904

3^2 + 26^2 + 76^2 = 4^2 + 19^2 + 78^2 = 6461
3 x 26 x 76 = 4 x 19 x 78 = 5928

7^2 + 22^2 + 39^2 = 11^2 + 13^2 + 42^2 = 2054
7 x 22 x 39 = 11 x 13 x 42 = 6006

1^2 + 74^2 + 84^2 = 4^2 + 14^2 + 111^2 = 12533
1 x 74 x 84 = 4 x 14 x 111 = 6216

4^2 + 32^2 + 49^2 = 7^2 + 16^2 + 56^2 = 3441
4 x 32 x 49 = 7 x 16 x 56 = 6272

3^2 + 45^2 + 49^2 = 5^2 + 21^2 + 63^2 = 4435
3 x 45 x 49 = 5 x 21 x 63 = 6615

4^2 + 40^2 + 42^2 = 10^2 + 12^2 + 56^2 = 3380
4 x 40 x 42 = 10 x 12 x 56 = 6720

8^2 + 26^2 + 35^2 = 13^2 + 14^2 + 40^2 = 1965
8 x 26 x 35 = 13 x 14 x 40 = 7280

7^2 + 30^2 + 36^2 = 9^2 + 20^2 + 42^2 = 2245
7 x 30 x 36 = 9 x 20 x 42 = 7560

4^2 + 40^2 + 48^2 = 8^2 + 16^2 + 60^2 = 3920
4 x 40 x 48 = 8 x 16 x 60 = 7680
6^2 + 32^2 + 40^2 = 10^2 + 16^2 + 48^2 = 2660
6 x 32 x 40 = 10 x 16 x 48 = 7680

5^2 + 33^2 + 49^2 = 7^2 + 21^2 + 55^2 = 3515
5 x 33 x 49 = 7 x 21 x 55 = 8085

4^2 + 23^2 + 90^2 = 9^2 + 10^2 + 92^2 = 8645
4 x 23 x 90 = 9 x 10 x 92 = 8280

1^2 + 50^2 + 168^2 = 6^2 + 8^2 + 175^2 = 30725
1 x 50 x 168 = 6 x 8 x 175 = 8400

1^2 + 86^2 + 102^2 = 2^2 + 34^2 + 129^2 = 17801
1 x 86 x 102 = 2 x 34 x 129 = 8772

1^2 + 62^2 + 145^2 = 2^2 + 29^2 + 155^2 = 24870
1 x 62 x 145 = 2 x 29 x 155 = 8990

7^2 + 24^2 + 54^2 = 9^2 + 18^2 + 56^2 = 3541
7 x 24 x 54 = 9 x 18 x 56 = 9072

1^2 + 66^2 + 140^2 = 4^2 + 15^2 + 154^2 = 23957
1 x 66 x 140 = 4 x 15 x 154 = 9240
3^2 + 44^2 + 70^2 = 4^2 + 30^2 + 77^2 = 6845
3 x 44 x 70 = 4 x 30 x 77 = 9240
4^2 + 42^2 + 55^2 = 7^2 + 20^2 + 66^2 = 4805
4 x 42 x 55 = 7 x 20 x 66 = 9240
5^2 + 28^2 + 66^2 = 11^2 + 12^2 + 70^2 = 5165
5 x 28 x 66 = 11 x 12 x 70 = 9240

2^2 + 56^2 + 84^2 = 4^2 + 24^2 + 98^2 = 10196
2 x 56 x 84 = 4 x 24 x 98 = 9408

7^2 + 35^2 + 39^2 = 13^2 + 15^2 + 49^2 = 2795
7 x 35 x 39 = 13 x 15 x 49 = 9555

2^2 + 39^2 + 125^2 = 5^2 + 15^2 + 130^2 = 17150
2 x 39 x 125 = 5 x 15 x 130 = 9750
3^2 + 50^2 + 65^2 = 5^2 + 25^2 + 78^2 = 6734
3 x 50 x 65 = 5 x 25 x 78 = 9750

4^2 + 37^2 + 66^2 = 11^2 + 12^2 + 74^2 = 5741
4 x 37 x 66 = 11 x 12 x 74 = 9768

1^2 + 72^2 + 136^2 = 4^2 + 16^2 + 153^2 = 23681
1 x 72 x 136 = 4 x 16 x 153 = 9792

4^2 + 25^2 + 98^2 = 7^2 + 14^2 + 100^2 = 10245
4 x 25 x 98 = 7 x 14 x 100 = 9800

Paul.

• benvitalis says:

Thanks. I posted parametric solutions