Jacobi's two-square theorem states that the number of representations of a positive integer k as a sum of two squares, counting order and sign, is 4 times the surplus of positive divisors of k congruent to 1 modulo 4 over those congruent to 3 modulo 4. In this paper we give numerous identities, each of which yields an analogue of Jacobi's result. Our identities are drawn from a much larger list, and involve polygonal numbers. The formula for the n th k−gonal number is
Simson's identity is a well-known Fibonacci identity in which the difference of certain order 2 products has a particularly pleasing form. Other old and beautiful identities of a similar nature are attributed to Catalan, Gelin and Cesàro, and Tagiuri. Catalan's identity can be described as a family of product difference Fibonacci identities of order 2 with 1 parameter. In Section 2 of this paper we present four families of product difference Fibonacci identities that involve higher order products. Being self-dual, each of these families may be regarded as a higher order analogue of Catalan's identity. We also state two conjectures that give the form of similar families of arbitrary order. In the final section we give other interesting product difference Fibonacci identities.
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