Some conjectures in elementary number theory (original) (raw)
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Abstract factorial functions and their applications
2007
A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive integers we construct a "factorial set" with which one may define a multitude of abstract factorials on X. We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fibonacci numbers, and highly composite numbers among other sets of integers. One of our results states that given any abstract factorial the series of reciprocals of its factorials always converges to an irrational number. Thus, for example, for any positive integer k the series of the reciprocals of the k-th powers of the cumulative product of the divisors of the numbers from 1 to n is irrational.
Abstract factorials of arbitrary sets of integers
2007
A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X ⊆ Z + we construct a "factorial set" with which one may define a multitude of abstract factorials on X. We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fibonacci numbers, and highly composite numbers among other sets of integers. One of our results states that given any abstract factorial the series of reciprocals of its factorials always converges to an irrational number. Thus, for example, for any positive integer k the series of the reciprocals of the k-th powers of the cumulative product of the divisors of the numbers from 1 to n is irrational. n j=1 σ k (j) ,
arXiv (Cornell University), 2007
A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X ⊆ Z + we construct a "factorial set" with which one may define a multitude of abstract factorials on X. We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fibonacci numbers, and highly composite numbers among other sets of integers. One of our results states that given any abstract factorial the series of reciprocals of its factorials always converges to an irrational number. Thus, for example, for any positive integer k the series of the reciprocals of the k-th powers of the cumulative product of the divisors of the numbers from 1 to n is irrational. n j=1 σ k (j) ,
Some conjectures in elementary number theory, II
2014
We announce a number of conjectures associated with and arising from a study of primes and irrationals in R. All are supported by numerical verification to the extent possible. This is an unpublished updated version as of August 13, 2014.
Notes on Number Theory and Discrete Mathematics
2003
1. The Smarandache, Pseudo-Smarandache, resp. Smarandache-simple functions are defined as ([7J, [6]) S{n) = min{rn EN: nlm!}, Z(n) = min {m.E N: nl m{n~ + 1)} , 5p (n) = min{m EN: p"lm!} for fixed primes p. The duals of Sand Z have been studied e.g. in (2], [5J, [6]: 5.(n) = max{m EN: m!ln}, { m(rn+1) } Z.(n) = max mEN: 2
New Equivalents of Kurepa’s Hypothesis for Left Factorial
Axioms
Kurepa’s hypothesis for the left factorial has been an unsolved problem for more than 50 years. In this paper, we have proposed new equivalents for Kurepa’s hypothesis for the left factorial. The connection between the left factorial and the continued fractions is given. The new equivalent based on the properties of the integer part of real numbers is proven. Moreover, a new equivalent based on the properties of two well-known sequences is given. A new representation of the left factorial is listed. Since derangement numbers are closely related to Kurepa’s hypothesis, we made some notes about the derangement numbers and defined a new sequence of natural numbers based on the derangement numbers. In this paper, we indicate a possible direction for further research through solving quadratic equations.
The exponents in the prime decomposition of factorials
Archiv der Mathematik, 2016
Let νp(n) be the exponent of p in the prime decomposition of n. We show that for different primes p, q satisfying some mild constraints the integers νp(n!) and νq(n!) cannot both be of a rather special form.
On the Parity of Exponents in the Factorization ofn!
Journal of Number Theory, 1997
It is shown that, for any k, there exist infinitely many positive integers n such that in the prime power factorization of n!, all first k primes appear to even exponents. This answers a question of Erdo s and Graham (``Old and New Problems and Results in Combinatorial Number Theory,'' L'Enseignement Mathe matique, Imprimerie Kundia, Geneva, 1980). A few generalizations are provided as well. 1997 Academic Press Question. Does there exist, for every fixed k, some n>1 with all the exponents _ 1 (n), _ 2 (n), ..., _ k (n) even? Our first result answers this question in the affirmative.
On the decomposition of n! into primes
Abstract. In this note, we make explicit approximation of the average of prime powers in the decomposition of n!. Then we find the order of geometric and harmonic means of such powers. ... 1.1. Approximate Formula for the Function Υ(n). First, we note that integrating by
On the Generalization of Factoriangular Numbers
Asian Research Journal of Mathematics
A factoriangular number is a sum of a factorial and its corresponding triangular number. This paper presents some forms of the generalization of factoriangular numbers. One generalization is the \(n^{(m)}\) -factoriangular number which is of the form \((n!)^{m}\) + \(S_m(n)\), where \((n!)^{m}\) is the \(m\)th power of the factorial of \(n\) and \(S_m(n)\) is the sum of the \(m\)-powers of \(n\). This generalized form is explored for the different values of the natural number \(m\). The investigation results to some interesting proofs of theorems related thereto. Two important formulas were generated for \((n)^{m}\) -factoriangular number: \(Ft_{n^{(m)}}\) = \(Ft_{n^{(2k)}}\) = \((n!)^{2k}\) + \(2n+1\over2k+1\)\([n^{2k-2}+P(n^{2k-3})]T_n\) for even \(m=2k\), and \(Ft_{n^{(m)}}\) = \(Ft_{n^{(2k+1)}}\) = \((n!)^{2k+1}\) + \(n(n+1)\over k+1\)\([n^{2k-2}+P(n^{2k-3})]T_n\) for odd \(m=2k+1\)