Cassini's identity for k-bonacci numbers
arXiv:2606.0322816.3
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Provides a definitive solution to a decades-old open problem in number theory concerning k-bonacci numbers.
The authors prove Cassini's identity for k-bonacci numbers, along with generalizations of Catalan's and Vajda's identities, providing a complete and simple proof for a long-standing open problem.
Efforts have been made to extend Cassini's identity (also known as Simson's identity) to the k-step or k-bonacci numbers for decades. These efforts have lacked both completeness of result and simplicity of proof, and this question remains open and relevant. In this note, we offer a definitive solution as well as the generalization of both Catalan's and Vajda's identities.