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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.
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(x) ..."
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Cited by 5 (5 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}.
A SEARCH FOR FIBONACCIWIEFERICH AND WOLSTENHOLME PRIMES
"... Abstract. Aprimepis called a FibonacciWieferich prime if F p p− ( ) ≡ 0 5 (mod p2), where Fn is the nth Fibonacci number. We report that there exist no such primes p <2 × 1014. A prime p is called a Wolstenholme prime if �2p−1 � ≡ 1(modp4). To date the only known Wolstenholme primes are p−1 16843 ..."
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Abstract. Aprimepis called a FibonacciWieferich prime if F p p− ( ) ≡ 0 5 (mod p2), where Fn is the nth Fibonacci number. We report that there exist no such primes p <2 × 1014. A prime p is called a Wolstenholme prime if �2p−1 � ≡ 1(modp4). To date the only known Wolstenholme primes are p−1 16843 and 2124679. We report that there exist no new Wolstenholme primes p<109. Wolstenholme, in 1862, proved that �2p−1 � ≡ 1(modp3)forall p−1 primes p ≥ 5. It is estimated by a heuristic argument that the “probability” that p is FibonacciWieferich (independently: that p is Wolstenholme) is about 1/p. We provide some statistical data relevant to occurrences of small values of the FibonacciWieferich quotient F p p− ()/p modulo p. 5 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 4 (3 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.
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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Cited by 2 (2 self)
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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
A COVERING SYSTEM WITH LEAST MODULUS 25
, 2008
"... Abstract. A collection of congruences with distinct moduli, each greater than 1, such that each integer satisfies at least one of the congruences, is said to be a covering system. A famous conjecture of Erdös from 1950 states that the least modulus of a covering system can be arbitrarily large. This ..."
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Abstract. A collection of congruences with distinct moduli, each greater than 1, such that each integer satisfies at least one of the congruences, is said to be a covering system. A famous conjecture of Erdös from 1950 states that the least modulus of a covering system can be arbitrarily large. This conjecture remains open and, in its full strength, appears at present to be unattackable. Most of the effort in this direction has been aimed at explicitly constructing covering systems with large least modulus. Improving upon previous results of Churchhouse, Krukenberg, Choi, and Morikawa, we construct a covering system with least modulus 25. The construction involves a largescale computer search, in conjunction with two general results that considerably reduce the complexity of the search. 1.
PRIMITIVE REPRESENTATIONS OF INTEGERS BY x 3 + y 3 + 2z 3
, 2010
"... Abstract. A wellknown open problem is to show that the cubic form x 3 + y 3 + 2z 3 represents all integers. An obvious variant of this problem is whether every integer can be primitively represented by x 3 + y 3 + 2z 3. In other words, given an integer n, are there coprime integers x, y, z such tha ..."
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Abstract. A wellknown open problem is to show that the cubic form x 3 + y 3 + 2z 3 represents all integers. An obvious variant of this problem is whether every integer can be primitively represented by x 3 + y 3 + 2z 3. In other words, given an integer n, are there coprime integers x, y, z such that x 3 + y 3 + 2z 3 = n? In this note we answer this variant question negatively. Indeed, we use cubic reciprocity to show that for every integral solution to x 3 + y 3 + 2z 3 = 8 m, the unknowns x, y, z are divisible by 2 m. Guy [2, Problem D5] asks if every number is the sum of four cubes with two of them equal. In other words, can every integer be represented by the cubic form x 3 + y 3 + 2z 3. Guy notes that this has been shown to be the case for all positive integers < 1000 except for 148, 671, 788 which are still open. An obvious variant of this problem is whether every integer can be primitively represented by x 3 + y 3 + 2z 3. In other words, given an integer n, are there coprime integers x, y, z such that x 3 + y 3 + 2z 3 = n? In this note we answer this question negatively, as the following theorem shows. Theorem 1. Let m ≥ 1. If x, y, z are integers satisfying (1) x 3 + y 3 + 2z 3 = 8 m then x, y and z are divisible by 2 m. The proof, given in Section 2, uses cubic reciprocity. The use of cubic reciprocity is inspired by Cassels ’ [1] where he shows that any integral solution to x 3 +y 3 +z 3 = 3 satisfies x ≡ y ≡ z (mod 9). We summarize what we need from cubic reciprocity
CYCLES AND FIXED POINTS OF HAPPY FUNCTIONS
, 2009
"... Let N = {1, 2, 3, · · · } denote the natural numbers. Given integers e ≥ 1 and b ≥ 2, let x = ∑ n i=0 aib i with 0 ≤ ai ≤ b − 1 (thus ai are the digits of x in base b). We define the happy function Se,b: N − → N by Se,b(x) = A positive integer x is then said to be (e, b)happy if Sr e,b (x) = 1 ..."
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Let N = {1, 2, 3, · · · } denote the natural numbers. Given integers e ≥ 1 and b ≥ 2, let x = ∑ n i=0 aib i with 0 ≤ ai ≤ b − 1 (thus ai are the digits of x in base b). We define the happy function Se,b: N − → N by Se,b(x) = A positive integer x is then said to be (e, b)happy if Sr e,b (x) = 1 for some r ≥ 0, otherwise we say it is (e, b)unhappy. In this paper we investigate the cycles and fixed points of the happy functions Se,b. We give an upper bound for the size of elements belonging to the cycles of Se,b. We also prove that the number of fixed points of S2,b is