3^3 + 4^3 + 5^3 = 6^3 Can anyone else think of another nontrivial example?

Question

Consider W^3 + X^3 + Y^3 = Z^3 where W, X, Y, and Z are positive integers.  What is another example of (W, X, Y, Z) satisfying the equation without (W, X, Y, Z) being a integer multiple of (3, 4, 5, 6)?  For example, I am not interested in
6^3 + 8^3 + 10^3 = 12^3 because we already know that the integer multiples of (3, 4, 5, 6) work.

lou h · Accepted Answer

5³ + 163³ + 164³ = 206³ 

7³ + 54³ + 57³ = 70³ 

23³ + 97³ + 86³ = 116³ 
 
3³ + 36³ + 37³ = 46³

.....

Saludos.

Pranil · Answer

3^2 + 4^2 = 5^2
10^2 + 11^2 + 12^2 = 13^2 + 14^2
 21^2 + 22^2 + 23^2 + 24^2 +  = 25^2 + 26^2 + 27^2 
36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2

Mein Hoon Na · Answer

the general solution of

x^3+y^3 = z^3 + w^3

is x = 9t^2-1
y = 1
z= 9t^4
w = 3t-9t^4 = 3t(1-9t^3)
by choosing t > 1 we can make x,y z as positive and w -ve so get of the form you require
t =1 gives x = 8, y= 1, z = 9 w = -6 or
8^3+6^2+ 1^3 = 9^3
(if you require the solution steps please let me know)

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3^3 + 4^3 + 5^3 = 6^3 Can anyone else think of another nontrivial example?

3 Answers