## a^2 + a*b + b^2 = c^2 + c*d + d^2

$N^2 \; = \; a^2 \; + \; a \, b \; + \; b^2$

parametric solutions:

$N \; = \; u^2 \; + \; u \, v \; + \; v^2$
$a \; = \; u^2 \; - \; v^2$
$b \; = \; 2 \, u \, v \; + \; v^2$

$a^2 \; + \; a \, b \; + \; b^2$
$= \; (u^2 - v^2)^2 \; + \; (u^2 - v^2)(2 \, u \, v + v^2) \; + \; (2 \, u \, v + v^2)^2$
$= \; u^4 \; + \; 2 \, u^3 \, v \; + \; 3 \, u^2 \, v^2 \; + \; 2 \, u \, v^3 \; + \; v^4$
$= \; (u^2 + u \, v + v^2)^2$
$= \; N^2$

$(a, b) = 1$

Here are the first few solutions:

$7^2 \; = \; 3^2 \; + \; 15 \; + \; 5^2$
$13^2 \; = \; 8^2 \; + \; 56 \; + \; 7^2$
$19^2 \; = \; 5^2 \; + \; 80 \; + \; 16^2$
$21^2 \; = \; 15^2 \; + \; 135 \; + \; 9^2$
$31^2 \; = \; 24^2 \; + \; 264 \; + \; 11^2$
$37^2 \; = \; 7^2 \; + \; 231 \; + \; 33^2$
$39^2 \; = \; 21^2 \; + \; 504 \; + \; 24^2$
$49^2 \; = \; 16^2 \; + \; 624 \; + \; 39^2$
$61^2 \; = \; 9^2 \; + \; 504 \; + \; 56^2$

Can you find distinct integers   $a, b, c, d$   that satisfy:

(1)   $a^2 \; + \; a \, b \; + \; b^2 \; = \; c^2 \; + \; c \, d \; + \; d^2$

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

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## About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

### 2 Responses to a^2 + a*b + b^2 = c^2 + c*d + d^2

1. paul says:

There are many solutions this is a sample with b<a<=30
For part a

6^2 + (6 x 5) + 5^2 = 91
9^2 + (9 x 1) + 1^2 = 91

9^2 + (9 x 4) + 4^2 = 133
11^2 + (11 x 1) + 1^2 = 133

11^2 + (11 x 8) + 8^2 = 273
16^2 + (16 x 1) + 1^2 = 273

14^2 + (14 x 7) + 7^2 = 343
18^2 + (18 x 1) + 1^2 = 343

13^2 + (13 x 5) + 5^2 = 259
15^2 + (15 x 2) + 2^2 = 259

12^2 + (12 x 10) + 10^2 = 364
18^2 + (18 x 2) + 2^2 = 364

14^2 + (14 x 9) + 9^2 = 403
19^2 + (19 x 2) + 2^2 = 403

9^2 + (9 x 8) + 8^2 = 217
13^2 + (13 x 3) + 3^2 = 217

11^2 + (11 x 7) + 7^2 = 247
14^2 + (14 x 3) + 3^2 = 247

17^2 + (17 x 6) + 6^2 = 427
19^2 + (19 x 3) + 3^2 = 427

13^2 + (13 x 12) + 12^2 = 469
20^2 + (20 x 3) + 3^2 = 469

11^2 + (11 x 9) + 9^2 = 301
15^2 + (15 x 4) + 4^2 = 301

13^2 + (13 x 10) + 10^2 = 399
17^2 + (17 x 5) + 5^2 = 399

16^2 + (16 x 9) + 9^2 = 481
19^2 + (19 x 5) + 5^2 = 481

15^2 + (15 x 11) + 11^2 = 511
19^2 + (19 x 6) + 6^2 = 511

15^2 + (15 x 13) + 13^2 = 589
20^2 + (20 x 7) + 7^2 = 589

and a solution with 6 sets b < a <= 109

66^2 + (66 x 61) + 61^2 = 12103
77^2 + (77 x 49) + 49^2 = 12103
89^2 + (89 x 34) + 34^2 = 12103
94^2 + (94 x 27) + 27^2 = 12103
98^2 + (98 x 21) + 21^2 = 12103
109^2 + (109 x 2) + 2^2 = 12103

Part b
b < a <= 109

22^4 + (22^2 x 17^2) + 17^4 = 457653
26^4 + (26^2 x 1^2) + 1^4 = 457653

80^4 + (80^2 x 47^2) + 47^4 = 59977281
88^4 + (88^2 x 1^2) + 1^4 = 59977281

25^4 + (25^2 x 22^2) + 22^4 = 927381
31^4 + (31^2 x 2^2) + 2^4 = 927381

44^4 + (44^2 x 34^2) + 34^4 = 7322448
52^4 + (52^2 x 2^2) + 2^4 = 7322448

66^4 + (66^2 x 51^2) + 51^4 = 37069893
78^4 + (78^2 x 3^2) + 3^4 = 37069893

48^4 + (48^2 x 23^2) + 23^4 = 6807073
51^4 + (51^2 x 4^2) + 4^4 = 6807073

50^4 + (50^2 x 44^2) + 44^4 = 14838096
62^4 + (62^2 x 4^2) + 4^4 = 14838096

88^4 + (88^2 x 68^2) + 68^4 = 117159168
104^4 + (104^2 x 4^2) + 4^4 = 117159168

55^4 + (55^2 x 53^2) + 53^4 = 25538331
71^4 + (71^2 x 5^2) + 5^4 = 25538331

75^4 + (75^2 x 66^2) + 66^4 = 75117861
93^4 + (93^2 x 6^2) + 6^4 = 75117861

55^4 + (55^2 x 16^2) + 16^4 = 9990561
56^4 + (56^2 x 7^2) + 7^4 = 9990561

20^4 + (20^2 x 19^2) + 19^4 = 434721
25^4 + (25^2 x 8^2) + 8^4 = 434721

44^4 + (44^2 x 15^2) + 15^4 = 4234321
45^4 + (45^2 x 8^2) + 8^4 = 4234321

96^4 + (96^2 x 46^2) + 46^4 = 108913168
102^4 + (102^2 x 8^2) + 8^4 = 108913168

17^4 + (17^2 x 16^2) + 16^4 = 223041
20^4 + (20^2 x 11^2) + 11^4 = 223041

40^4 + (40^2 x 38^2) + 38^4 = 6955536
50^4 + (50^2 x 16^2) + 16^4 = 6955536

88^4 + (88^2 x 30^2) + 30^4 = 67749136
90^4 + (90^2 x 16^2) + 16^4 = 67749136

83^4 + (83^2 x 64^2) + 64^4 = 92452881
97^4 + (97^2 x 20^2) + 20^4 = 92452881

34^4 + (34^2 x 32^2) + 32^4 = 3568656
40^4 + (40^2 x 22^2) + 22^4 = 3568656

60^4 + (60^2 x 57^2) + 57^4 = 35212401
75^4 + (75^2 x 24^2) + 24^4 = 35212401

80^4 + (80^2 x 76^2) + 76^4 = 111288576
100^4 + (100^2 x 32^2) + 32^4 = 111288576

51^4 + (51^2 x 48^2) + 48^4 = 18066321
60^4 + (60^2 x 33^2) + 33^4 = 18066321

68^4 + (68^2 x 64^2) + 64^4 = 57098496
80^4 + (80^2 x 44^2) + 44^4 = 57098496

85^4 + (85^2 x 80^2) + 80^4 = 139400625
100^4 + (100^2 x 55^2) + 55^4 = 139400625

Paul.

• benvitalis says:

Nice! These will appear in equations of the form
a^n + b^n + c^n = d^n + e^n + f^n, n = 2, 4
I’ll use them in my next blog