For m >= 2, the equation (L_m)^n + (L_m)^(n+k) = L_r has no positive integer solutions in r, m, n, and k.
Fibonacci Q.45, 98–103 (2007)
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On a Diophantine Equation Involving Lucas Numbers
For m >= 2, the equation (L_m)^n + (L_m)^(n+k) = L_r has no positive integer solutions in r, m, n, and k.