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SOME SOLVED AND UNSOLVED PROBLEMS IN COMBINATORIAL NUMBER THEORY, II
"... In an earlier paper [9], the authors discussed some solved and unsolved problems in combinatorial number theory. First we will give an update of some of these problems. In the remaining part of this paper we will discuss some further problems of the two authors. ..."
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In an earlier paper [9], the authors discussed some solved and unsolved problems in combinatorial number theory. First we will give an update of some of these problems. In the remaining part of this paper we will discuss some further problems of the two authors.
SETS OF MONOTONICITY FOR EULER’S TOTIENT FUNCTION
"... We study subsets of [1, x] on which the Euler ϕfunction is monotone (nondecreasing or nonincreasing). For example, we show that for any ɛ> 0, every such subset has size < ɛx, once x> x0(ɛ). This confirms a conjecture of the second author. ..."
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We study subsets of [1, x] on which the Euler ϕfunction is monotone (nondecreasing or nonincreasing). For example, we show that for any ɛ> 0, every such subset has size < ɛx, once x> x0(ɛ). This confirms a conjecture of the second author.
THE DISTRIBUTION OF VALUES OF A CERTAIN CLASS OF ARITHMETIC FUNCTIONS AT CONSECUTIVE INTEGERS
 COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 51. NUMBER THEORY, BUDAPEST (HUNGARY)
, 1987
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LINEAR EQUATIONS WITH THE EULER TOTIENT FUNCTION
"... Abstract. In this paper, we investigate linear relations among the Euler function of nearby integers. In particular, we study those positive integers n such that φ(n) = φ(n − 1) + φ(n − 2), where φ is the Euler function. We prove that they form a set of asymptotic density zero. We also show that t ..."
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Abstract. In this paper, we investigate linear relations among the Euler function of nearby integers. In particular, we study those positive integers n such that φ(n) = φ(n − 1) + φ(n − 2), where φ is the Euler function. We prove that they form a set of asymptotic density zero. We also show that the sum of the reciprocals of the prime values of n with the above property is a convergent series.
ON THE SOLUTIONS TO (n) = (n + k)
"... Abstract. We study the number and nature of solutions of the equation (n) = (n + k), where denotes Euler's phifunction. 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 f ..."
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Abstract. We study the number and nature of solutions of the equation (n) = (n + k), where denotes Euler's phifunction. 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 ktuples conjecture. We also show that the prime ktuples conjecture implies that there are arbitrarily long arithmetic progressions of equal values. 1.
ON THE DIFFERENCE OF VALUES OF THE KERNEL FUNCTION AT CONSECUTIVE INTEGERS
, 2003
"... For each positive integer n, setγ(n) = ∏ pn p. Given a fixed integer k ≠ ±1, we establish that if the ABCconjecture holds, then the equation γ(n+1)−γ(n) = k has only finitely many solutions. In the particular cases k =±1, we provide a large family of solutions for each of the corresponding equat ..."
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For each positive integer n, setγ(n) = ∏ pn p. Given a fixed integer k ≠ ±1, we establish that if the ABCconjecture holds, then the equation γ(n+1)−γ(n) = k has only finitely many solutions. In the particular cases k =±1, we provide a large family of solutions for each of the corresponding equations.