Bruce Brandt

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Number theory; Congruences and residues


Let n and k be arbitrary positive integers. We will tend to be concerned with small k and with n which are several times k!. I stated two conditions on (n, k) in a previous paper; in this paper I restate them and further explore them. In particular, it is proven that if n is the least number satisfying Condition 1 for a certain k, then the least number for k + l must be at least 2n + l. Condition 1 and Condition 2 are rephrased graph-theoretically. A heuristic explanation for why the quadratic. residues tend to satisfy Condition 2b is given. A conjecture characterizes n and k which satisfy Condition 2b when IS a prime of the form 4m + l and Q are the quadratic residues. The case of the quadratic residues or non-residues with zero appended to them is discussed.

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