1,446 research outputs found

    On Egyptian fractions

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    We find a polynomial in three variables whose values at nonnegative integers satisfy the Erd\H{o}s-Straus Conjecture. Although the perfect squares are not covered by these values, it allows us to prove that there are arbitrarily long sequence of consecutive numbers satisfying the Erd\H{o}s-Straus Conjecture. We conjecture that the values of this polynomial include all the prime numbers of the form 4q+54q+5, which is checked up to 101410^{14}. A greedy-type algorithm to find an Erd\H{o}s-Straus decomposition is also given; the convergence of this algorithm is proved for a wide class of numbers. Combining this algorithm with the mentioned polynomial we verify that all the natural numbers nn, 2n2×10142\le n\le 2\times 10^{14}, satisfy the Ed\H{o}s-Straus Conjecture.Comment: 24 pages. Summit to a journal. This is a new version of our old paper. We include new results, see the new abstrac

    On the Erdos-Straus conjecture

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    Paul Erdos conjectured that for every n in N, n>1, there exist a, b, c natural numbers, not necessarily distinct, so that 4/n=1/a+1/b+1/c (see \cite{rg}). In this paper we prove an extension of Mordell's theorem and formulate a conjecture which is stronger than Erdos' conjecture.Comment: 9 pages, no figure

    Unit Fractions in Norm-Euclidean Rings of Integers

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    In this paper we make a Gaussian integer version of the Erd\H{o}s-Straus conjecture and we solve the Erd\H{o}s-Straus diophantine equation over the rings of integers of norm-Euclidean quadratic fields

    The Herzog-Sch\"onheim Conjecture for small groups and harmonic subgroups

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    We prove that the Herzog-Sch\"onheim Conjecture holds for any group GG of order smaller than 14401440. In other words we show that in any non-trivial coset partition {giUi}i=1n\{g_i U_i\}_{i=1}^n of GG there exist distinct 1i,jn1 \leq i, j \leq n such that [G:Ui]=[G:Uj][G:U_i]=[G:U_j]. We also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if U1U_1,...,UnU_n are subgroups of GG which have pairwise trivially intersecting cosets and n4n \leq 4 then [G:U1][G:U_1],...,[G:Un][G:U_n] are harmonic integers.Comment: 16 page

    The Erd\H{o}s-Straus conjecture New modular equations and checking up to N=1017N=10^{17}

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    In 1999 Allan Swett checked (in 150 hours) the Erd\H{o}s-Straus conjecture up to N=1014N=10^{14} with a sieve based on a single modular equation. After having proved the existence of a "complete" set of seven modular equations (including three new ones), this paper offers an optimized sieve based on these equations. A program written in C++ (and given elsewhere) allows then to make a checking whose running time, on a typical computer, range from few minutes for N=1014N=10^{14} to about 16 hours for N=1017N=10^{17}.Comment: 13 pages, 1 version fran\c{c}aise, 1 C++ progra

    Dense Egyptian Fractions

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    Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest demoninator. We show that for every positive rational there exist Egyptian fractions whose largest denominator is at most N and whose denominators form a positive proportion of the integers up to N, for sufficiently large N; furthermore, the proportion is within a small factor of best possible.Comment: 16 pages; to appear in Trans. Amer. Math. So

    Factorization length distribution for affine semigroups I: numerical semigroups with three generators

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    Most factorization invariants in the literature extract extremal factorization behavior, such as the maximum and minimum factorization lengths. Invariants of intermediate size, such as the mean, median, and mode factorization lengths are more subtle. We use techniques from analysis and probability to describe the asymptotic behavior of these invariants. Surprisingly, the asymptotic median factorization length is described by a number that is usually irrational.Comment: 17 page

    Egyptian Fractions and Prime Power Divisors

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    From varying Egyptian fraction equations we obtain generalizations of primary pseudoperfect numbers and Giuga numbers which we call prime power psuedoperfect numbers and prime power Giuga numbers respectively. We show that a sequence of Amarnath Murthy in the OEIS is a subsequence of the sequence of prime power psuedoperfect numbers. Prime factorization conditions sufficient to imply a number is a prime power pseudoperfect number or a prime power Giuga number are given. The conditions on prime factorizations naturally give rise to a generalization of Fermat primes which we call extended Fermat primes

    Egyptian Fractions with odd denominators

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    The number of solutions of the diophantine equation i=1k1xi=1,\sum_{i=1}^k \frac{1}{x_i}=1, in particular when the xix_i are distinct odd positive integers is investigated. The number of solutions S(k)S(k) in this case is, for odd kk: exp(exp(c1klogk))S(k)exp(exp(c2k))\exp \left( \exp \left( c_1\, \frac{k}{\log k}\right)\right) \leq S(k) \leq \exp \left( \exp \left(c_2\, k \right)\right) with some positive constants c1c_1 and c2c_2. This improves upon an earlier lower bound of S(k)exp((1+o(1))log22k2)S(k) \geq \exp \left( (1+o(1))\frac{\log 2}{2} k^2\right)

    An ancient Egyptian problem:the diophantine equation 4/n=1/x+1/y+1/z, n>or=2

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    From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that has come down to us in the last 60 years, is known as the Erdos- Strauss conjecture which states that for each positive integer n>1; the fraction 4/n can be decomposed into a sum of three distinct unit fractions. Since 1950, a numberof partial results have been achieved, see references [1]- [8]; and also [10] and[11]. In this work we contribute four theorems. In Theorem 2, we prove that if the fraction 4/n is not equal to a sum of three distinct unit fractions, then each prime divisor of n; must be congruent to 1 modulo24. Moreover, if n contains a divisor noncongruent to 1mod24; then such a decomposition does exist and it is explicitly given.Comment: 9 pages, no figure
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