a^n + b^n + (a-b)^n = c^n + d^n + (c-d)^n, n = 2,4

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

= \; a^2 \; - \; a \, b \; + \; b^2 \; = \; c^2 \; - \; c \, d \; + \; d^2
or equivalently
a^2 \; + \; a \, b \; + \; b^2 \; = \; c^2 \; + \; c \, d \; + \; d^2

a^4 \; + \; b^4 \; + \; (a - b)^4 \; = \; c^4 \; + \; d^4 \; + \; (c - d)^4
2 \, a^4 - 4 \, a^3 \, b + 6 \, a^2 \, b^2 - 4 \, a \, b^3 + 2 \, b^4
= 2 \, c^4 - 4 \, c^3 \, d + 6 \, c^2 \, d^2 - 4 \, c \, d^3 + 2 \, d^4

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

(a^2 - a \, b + b^2)^2 \; = \; (c^2 - c \, d + d^2)^2
or equivalently
(a^2 + a \, b + b^2)^2 \; = \; (c^2 + c \, d + d^2)^2
 
 

17^2 + 1^2 + 16^2 \; = \; 19^2 + 11^2 + 8^2
17^4 + 1^4 + 16^4 \; = \; 19^4 + 11^4 + 8^4

22^2 + 1^2 + 23^2 \; = \; 26^2 + 13^2 + 13^2
22^4 + 1^4 + 23^4 \; = \; 26^4 + 13^4 + 13^4

27^2 + 3^2 + 30^2 \; = \; 33^2 + 15^2 + 18^2
27^4 + 3^4 + 30^4 \; = \; 33^4 + 15^4 + 18^4

32^2 + 5^2 + 37^2 \; = \; 40^2 + 17^2 + 23^2
32^4 + 5^4 + 37^4 \; = \; 40^4 + 17^4 + 23^4

37^2 + 7^2 + 44^2 \; = \; 47^2 + 19^2 + 28^2
37^4 + 7^4 + 44^4 \; = \; 47^4 + 19^4 + 28^4

42^2 + 9^2 + 51^2 \; = \; 54^2 + 21^2 + 33^2
42^4 + 9^4 + 51^4 \; = \; 54^4 + 21^4 + 33^4

47^2 + 11^2 + 58^2 \; = \; 61^2 + 23^2 + 38^2
47^4 + 11^4 + 58^4 \; = \; 61^4 + 23^4 + 38^4

52^2 + 13^2 + 65^2 \; = \; 68^2 + 25^2 + 43^2
52^4 + 13^4 + 65^4 \; = \; 68^4 + 25^4 + 43^4

57^2 + 15^2 + 72^2 \; = \; 75^2 + 27^2 + 48^2
57^4 + 15^4 + 72^4 \; = \; 75^4 + 27^4 + 48^4

62^2 + 17^2 + 79^2 \; = \; 82^2 + 29^2 + 53^2
62^4 + 17^4 + 79^4 \; = \; 82^4 + 29^4 + 53^4

67^2 + 19^2 + 86^2 \; = \; 89^2 + 31^2 + 58^2
67^4 + 19^4 + 86^4 \; = \; 89^4 + 31^4 + 58^4

72^2 + 21^2 + 93^2 \; = \; 96^2 + 33^2 + 63^2
72^4 + 21^4 + 93^4 \; = \; 96^4 + 33^4 + 63^4

77^2 + 23^2 + 100^2 \; = \; 103^2 + 35^2 + 68^2
77^4 + 23^4 + 100^4 \; = \; 103^4 + 35^4 + 68^4

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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