Jeffrey J. Holt

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We study the number and nature of solutions of the equation φ(n) = φ(n + k), where φ denotes Euler’s phi-function. We exhibit some families of solutions when k is even, and we conjecture an asymptotic formula for the number of solutions in this case. We show that our conjecture follows from a quantitative form of the prime k-tuples conjecture. We also show(More)
We consider the minimal number of solutions to (n) = (n + k) for xed values of k. Previous work of A. Schinzel concerning this problem is described, as is the outcome of recent computational searches conducted to extend Schinzel's results. Let k be a xed positive integer. In this note, we consider the minimum number of solutions to the equation (n) = (n +(More)
Masuyama in Sankhya 14(3):181–186, 1954 gave a method for addressing the boundary overlap problem that arises when sampling in a delineated study region. Here we propose a general sampling framework that allows one to iterate Masuyama’s method in a way that provides a smaller sampling variance. Results of simulations comparing the proposed method to(More)
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