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.

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