Powers as Fibonacci sums.

AbstractWe examine the equation for positive integers <italic>y</italic>, <italic>a</italic> ≥ 2 and <italic>k</italic> ≥ 3. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for <italic>y</italic> ≤ 25000 when <italic>k</italic> = 3, for <italic>y</italic> ≤ 1000 when <italic>k</italic> = 4, for <italic>y</italic> ≤ 40 when <italic>k</italic> = 5, and for <italic>y</italic> ≤ 3 when <italic>k</italic> = 6. [ABSTRACT FROM AUTHOR]