Unit Fractions and Fibonacci
Let s[j], j=0,1,2,... be a sequence of integers that satisfy the recurrence relation s[k] = s[k-1] + s[k-2] with arbitrary initial values s and s (Note that if s = 1 and s = 1, then it is the Fibonacci Sequence). Use spreadsheet software to observe that for any integers m, n with m>n we have
For example, setting s=s=1 and n=5, m=10 gives
1/(s.s) = 1/(s.s) + 1/(s.s) +1/(s.s) +1/(s.s) +1/(s.s) +1/(s.s) +1/(s.s)
In other words,
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How can we use this observation to represent any given unit fraction as sum of certain number of unit fractions (e.g. 1/a as sum of b unit fraction)?
How can we use above relation to generate integer solutions of the harmonic equation
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