Polynomials which take Gaussian integer values at Gaussian integers
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A factorial set for the Gaussian integers is a set G = {g1, g2 ... gn} of Gaussian integers such that G(z) = k (z - gk) gk takes Gaussian integer values at Gaussian integers. We characterize factorial sets and give a lower bound for maxz2= n G(z). It is conjectured that there are infinitely many factorial sets. A Gaussian integer valued polynomial (GIP) is a polynomial with the title property. A bound similar to the above is given for maxz2=n G(z) if G(z) is a GIP. There is a relation between factorial sets and testing for GIP's. We discuss this and close with some examples of factorial sets, and speculate on how to find more. 1977.