Joshua Harrington

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The authors investigate k-colorings of the positive integers ≤ n for which the triples (a, b, c), where a, b and c are positive integers with a2 + b2 = c2 and c ≤ n, satisfy the condition that a, b and c are colored differently. In particular, they establish for ε > 0 and n sufficiently large, if k ≥ √ 3 (1+ε) logn/ log logn , then a k-coloring exists such(More)
A unit x in a commutative ring R with identity is called exceptional if 1−x is also a unit in R. For any integer n ≥ 2, define φe(n) to be the number of exceptional units in the ring of integers modulo n. Following work of Shapiro, Mills, Catlin and Noe on iterations of Euler’s φ-function, we develop analogous results on iterations of the function φe, when(More)
Define a covering system (or covering) of the integers as a finite collection of congruences x ≡ aj (mod mj), with 1 ≤ j ≤ r, such that every integer satisfies at least one of these congruences. As an interesting application of coverings, W. Sierpiński [4] showed that there are odd positive integers k for which k · 2 + 1 is composite for all integers n ≥ 0.(More)
In 1960 Sierpiński proved that there are infinitely many odd positive integers k such that k · 2 + 1 is composite for all positive integers n. A polynomial variation of Sierpiński’s result has been investigated by several people. More specifically, the question has been asked, for which integers d does there exist a polynomial f(x) ∈ Z[x] with f(1) 6= −d(More)
In a recent article, Nowicki introduced the concept of a special number. Specifically, an integer d is called special if for every integer m there exist solutions in non-zero integers a, b, c to the equation a + b − dc = m. In this article we investigate pairs of integers (n, d), with n ≥ 2, such that for every integer m there exist units a, b, and c in Zn(More)
Usually, vehicles are equipped with Global Positioning System (GPS), which can provide its position estimation. However, GPS can become erroneous or unavailable in cases of some indoor scenarios, such as tunnels and dense urban areas where there is no straight visibility to satellites. In Vehicular Ad Hoc Networks (VANets), some critical applications such(More)
In this paper, we investigate arithmetic progressions in the polygonal numbers with a fixed number of sides. We first show that four-term arithmetic progressions cannot exist. We then describe explicitly how to find all three-term arithmetic progressions. Finally, we show that not only are there infinitely many three-term arithmetic progressions, but that(More)