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WOLSTENHOLME’S THEOREM: ITS GENERALIZATIONS AND EXTENSIONS IN THE LAST HUNDRED AND FIFTY YEARS (1862–2012)
, 2011
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COVERS OF THE INTEGERS WITH ODD MODULI AND THEIR APPLICATIONS TO THE FORMS x m − 2 n AND x 2 − F3n/2
"... Abstract. In this paper we construct a cover {as(mod ns)} k s=1 of Z with odd moduli such that there are distinct primes p1,...,pk dividing 2n1 −1,...,2nk − 1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmet ..."
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Abstract. In this paper we construct a cover {as(mod ns)} k s=1 of Z with odd moduli such that there are distinct primes p1,...,pk dividing 2n1 −1,...,2nk − 1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the mth powers of whose terms are never of the form 2n ± pa with a, n ∈{0, 1, 2,...} and p a prime. We also construct another cover of Z with odd moduli and use it to prove that x2 − F3n/2 has at least two distinct prime factors whenever n ∈{0, 1, 2,...} and x ≡ a (mod M), where {Fi}i�0 is the Fibonacci sequence, and a and M are suitable positive integers having 80 decimal digits. 1.
COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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Cited by 7 (6 self)
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
A CONNECTION BETWEEN COVERS OF THE INTEGERS AND UNIT FRACTIONS
 ADV. IN APPL. MATH. 38(2007), NO. 2, 267–274.
, 2007
"... For integers a and n> 0, let a(n) denote the residue class {x ∈ Z: x ≡ a (mod n)}. Let A be a collection {as(ns)} k s=1 of finitely many residue classes such that A covers all the integers at least m times but {as(ns)} k−1 s=1 does not. We show that if nk is a period of the covering function wA( ..."
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Cited by 3 (3 self)
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For integers a and n> 0, let a(n) denote the residue class {x ∈ Z: x ≡ a (mod n)}. Let A be a collection {as(ns)} k s=1 of finitely many residue classes such that A covers all the integers at least m times but {as(ns)} k−1 s=1 does not. We show that if nk is a period of the covering function wA(x) = {1 � s � k: x ∈ as(ns)}  then for any r = 0,..., nk−1 there are at least m integers in the form ∑ s∈I 1/ns − r/nk with I ⊆ {1,..., k − 1}.
Siegel’s lemma and SumDistinct sets
 Discrete Comput. Geom
, 2008
"... 1 Abstract. Let L(x) = a1x1 + a2x2 +... + anxn, n ≥ 2 be a linear form with integer coefficients a1, a2,...,an which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x) = 0, and the maximum norm x  is relatively small compared with the size of the coeffic ..."
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1 Abstract. Let L(x) = a1x1 + a2x2 +... + anxn, n ≥ 2 be a linear form with integer coefficients a1, a2,...,an which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x) = 0, and the maximum norm x  is relatively small compared with the size of the coefficients a1, a2,...,an. The main result of the paper asserts that there exist linearly independent vectors x1,...,xn−1 ∈ Z n such that L(xi) = 0, i = 1,...,n − 1 and where a = (a1, a2,...,an) and x1  · · · xn−1  < a σn = 2 π 0 sin t
Bennett’s Pillai theorem with fractional bases and negative exponents allowed
, 2013
"... In [3], Bennett proves Theorem A. If a, b, andc are positive integers with a, b ≥ 2, then the equation has at most two solutions in positive integers x and y and conjectures ..."
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In [3], Bennett proves Theorem A. If a, b, andc are positive integers with a, b ≥ 2, then the equation has at most two solutions in positive integers x and y and conjectures