Solving $a^3 + b^3 = N.c^3$

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Solving $a^3 + b^3 = N.c^3$

suppose $a^3 + b^3 = N.c^3$

there is solution for $x = 2+3w , w= ...-2,1,0,1,2...$

$N = x^3-1$ (for instance N = 7 , 65 , 124...)

$c = (1+x^3)/3$

$a = x(c-1)$

$b= x^3-c$

example: $w = 0 -> x = 2 , N = 2^3 - 1 = 7 , c = (1+2^3)/3 = 3 , a = 2(3-1) = 4 , b = 2^3-3 = 5$

$4^3 + 5^3 = 7.3^3$

example: $w = -2 -> x = -4 , N = -65 , c = -21 , a = 88 , b = -43$

$88^3 - 43^3 = (-65)(-21)^3 = 65.21^3$

Miguel Velilla