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Abstract

Let N be a positive integer, A be a subset of ℚ and α=α1α2A\{0,N}. N is called an α-Korselt number (equivalently α is said an N-Korselt base) if α 2 pα 1 divides α 2 Nα 1 for every prime divisor p of N. By the Korselt set of N over A, we mean the set AKS(N) of all αA\{0,N} such that N is an α-Korselt number.

In this paper we determine explicitly for a given prime number q and an integer l ∈ ℕ \{0, 1}, the set A-KS(ql) and we establish some connections between the ql -Korselt bases in ℚ and others in ℤ. The case of A=[1,1[ is studied where we prove that ([1,1[)-KS(ql) is empty if and only if l = 2.

Moreover, we show that each nonzero rational α is an N-Korselt base for infinitely many numbers N = ql where q is a prime number and l ∈ ℕ.

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Abstract  

We study the hybrid problem of Hua's theorem and the Piatetski-Shapiro prime number theorem, and obtain results in this direction of the nonhomogeneous case.

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Abstract  

We show that if A is a subset of {1, …, n} which has no pair of elements whose difference is equal to p − 1 with p a prime number, then the size of A is O(n(log log n)c(log log log log log n)) for some absolute c > 0.

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