# Zeckendorf's Theorem Using Indices in an Arithmetic Progression.

@article{Gilson2020ZeckendorfsTU, title={Zeckendorf's Theorem Using Indices in an Arithmetic Progression.}, author={Amelia Gilson and Hadley Killen and Tam{\'a}s Lengyel and Steven J. Miller and Nadia Razek and Joshua M. Siktar and Liza Sulkin}, journal={arXiv: Number Theory}, year={2020} }

Zeckendorf's Theorem states that any positive integer can be uniquely decomposed into a sum of distinct, non-adjacent Fibonacci numbers. There are many generalizations, including results on existence of decompositions using only even indexed Fibonacci numbers. We extend these further and prove that similar results hold when only using indices in a given arithmetic progression. As part of our proofs, we generate a range of new recurrences for the Fibonacci numbers that are of interest in their… CONTINUE READING

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