parametric solutions:

Here are the first few solutions:

Can you find distinct integers that satisfy:

(1)

(2)

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parametric solutions:

Here are the first few solutions:

Can you find distinct integers that satisfy:

(1)

(2)

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%d bloggers like this:

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.

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