1,446 research outputs found
On Egyptian fractions
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 , which is checked up to . 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 , , 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
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
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
We prove that the Herzog-Sch\"onheim Conjecture holds for any group of
order smaller than . In other words we show that in any non-trivial coset
partition of there exist distinct such that .
We also study interaction between the indices of subgroups having cosets with
pairwise trivial intersection and harmonic integers. We prove that if
,..., are subgroups of which have pairwise trivially intersecting
cosets and then ,..., are harmonic integers.Comment: 16 page
The Erd\H{o}s-Straus conjecture New modular equations and checking up to
In 1999 Allan Swett checked (in 150 hours) the Erd\H{o}s-Straus conjecture up
to 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
to about 16 hours for .Comment: 13 pages, 1 version fran\c{c}aise, 1 C++ progra
Dense Egyptian Fractions
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
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
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
The number of solutions of the diophantine equation in particular when the are distinct odd positive
integers is investigated. The number of solutions in this case is, for
odd : with some positive
constants and . This improves upon an earlier lower bound of
An ancient Egyptian problem:the diophantine equation 4/n=1/x+1/y+1/z, n>or=2
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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