Abstract
We define a Gauss factorial N n ! to be the product of all positive integers up to N that are relatively prime to n. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials \(\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\) for odd positive integers n. The case where n has exactly one prime factor of the form p≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by p with the property that the order of \(\frac{p-1}{4}!\) modulo p is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those n of the above form that satisfy \(\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}\). We also report on computations that were required in the process.
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Cosgrave, J.B., Dilcher, K. The Gauss–Wilson theorem for quarter-intervals. Acta Math Hung 142, 199–230 (2014). https://doi.org/10.1007/s10474-013-0357-1
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DOI: https://doi.org/10.1007/s10474-013-0357-1