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Representation of Integers as Sums of Fibonacci and Lucas Numbers
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Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.
Title: Representation of Integers as Sums of Fibonacci and Lucas Numbers
Description:
Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n.
Similar expressions are also dealt with for the case of Lucas numbers.
Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.
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