Repercussions of a Problem of Erdős and Ulam on Density Ideals

1990 ◽  
Vol 42 (5) ◽  
pp. 902-914 ◽  
Author(s):  
Winfried Just
Keyword(s):  
Boolean Algebra ◽  
Natural Number ◽  
Natural Numbers ◽  
Logarithmic Density ◽  
Density Zero ◽  
Image Position ◽  
The Ideal

By P(ω) we denote the Boolean algebra of all subsets of the set ω of natural numbers. We identify each natural number with the set of its predecessors and define: the ideal of sets of density zero, and the ideal of sets of logarithmic density zero.

Recursive Boolean algebras with recursive atoms

1981 ◽  
Vol 46 (3) ◽  
pp. 595-616 ◽  
Author(s):  
Jeffrey B. Remmel
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Isomorphism Type ◽  
Natural Numbers ◽  
Recursive Isomorphism ◽  
The Ideal

A Boolean algebra (henceforth abbreviated B.A.) is said to be recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Let denote the set of atoms of and denote the ideal generated by the atoms of . Given recursive B.A.s and , we write ≈ if is isomorphic to and ≈r if is recursively isomorphic to , i.e., if there is a partial recursive isomorphism from onto .Recursive B.A.s have been studied by several authors including Ershov [2], Fiener [3], [4], Goncharov [5], [6], [7], LaRoche [8], Nurtazin [7], and the author [10], [11]. This paper continues a study of the recursion theoretic relationships among , , and the recursive isomorphism type of a recursive B.A. we started in [11]. We refer the reader to [11] for any unexplained notation and definitions. In [11], we were mainly concerned with the possible recursion theoretic properties of the set of atoms in recursive B.A.s. We found that even if we insist that be recursive, there is considerable freedom for the properties of . For example, we showed that if is a recursive B.A. such that is recursive and is infinite, then (i) there exists a recursive B.A. such that and both and are recursive and (ii) for any nonzero r.e. degree δ, there exist recursive B.A.s , , … such that for each i, is of degree δ, is recursive, is immune if i is even and is not immune if i is odd, and no two B.A.s in the sequence are recursively isomorphic.

Note on the representation of integers as sums of certain powers

1969 ◽  
Vol 65 (2) ◽  
pp. 445-446 ◽  
Author(s):  
K. Thanigasalam
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Waring's Problem ◽  
Representation Of Integers ◽  
Image Position ◽  
Large Natural Number

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the formwhere x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).

On the Sieve of Eratosthenes

1987 ◽  
Vol 39 (5) ◽  
pp. 1107-1122 ◽  
Author(s):  
M. Ram Murty ◽  
S. Saradha
Keyword(s):  
Natural Number ◽  
Simple Proof ◽  
Classical Theorem ◽  
Natural Numbers ◽  
Normal Order ◽  
Prime Factors ◽  
Image Position

Let v(n) denote the number of distinct prime factors of a natural number n. A classical theorem of Hardy and Ramanujan states that the normal order of v(n) is log log n. That is, given any , the number of natural numbers not exceeding x which fail to satisfy the inequality1is o(x) as x → ∞. A very simple proof of this was subsequently given by Turán. He showed that2

On sums of two squarefull numbers

1994 ◽  
Vol 116 (1) ◽  
pp. 1-5 ◽  
Author(s):  
R. C. Baker ◽  
J. Brüdern
Keyword(s):  
Natural Number ◽  
Natural Numbers ◽  
Perfect Squares ◽  
Image Position

A natural number n is said to be squarefull if p|n implies p2|n for primes p. The set of all squarefull numbers is not much more dense in the natural numbers than the set of perfect squares but their additive properties may be rather different. We are more precise only in the case of sums of two such integers as this is the problem with which we are concerned here. Let U(x) be the number of integers not exceeding x and representable as the sum of two integer squares. Then, according to a theorem of Landau [4],as x tends to infinity.

Some formal relative consistency proofs

1953 ◽  
Vol 18 (2) ◽  
pp. 136-144 ◽  
Author(s):  
Robert McNaughton
Keyword(s):  
Number Theory ◽  
Natural Number ◽  
Natural Numbers ◽  
Quantification Theory ◽  
Usual Manner ◽  
Function Calculus ◽  
Image Position ◽  
Positive Integers

These systems are roughly natural number theory in, respectively, nth order function calculus, for all positive integers n. Each of these systems is expressed in the notation of the theory of types, having variables with type subscripts from 1 to n. Variables of type 1 stand for natural numbers, variables of type 2 stand for classes of natural numbers, etc. Primitive atomic wff's (well-formed formulas) of Tn are those of number theory in variables of type 1, and of the following kind for n > 1: xi ϵ yi+1. Other wff's are formed by truth functions and quantifiers in the usual manner. Quantification theory holds for all the variables of Tn. Tn has the axioms Z1 to Z9, which are, respectively, the nine axioms and axiom schemata for the system Z (natural number theory) on p. 371 of [1]. These axioms and axiom schemata contain only variables of type 1, except for the schemata Z2 and Z9, which are as follows:where ‘F(x1)’ can be any wff of Tn. Identity is primitive for variables of type 1; for variables of other types it is defined as follows:

A note on the representation of general recursive functions and the μ quantifier

1959 ◽  
Vol 55 (2) ◽  
pp. 145-148
Author(s):  
Alan Rose
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursive Functions ◽  
Image Position ◽  
Primitive Recursive

It has been shown that every general recursive function is definable by application of the five schemata for primitive recursive functions together with the schemasubject to the condition that, for each n–tuple of natural numbers x1,…, xn there exists a natural number xn+1 such that

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

The Range of a Gap Series

1975 ◽  
Vol 18 (5) ◽  
pp. 753-754
Author(s):  
J. S. Hwang
Keyword(s):  
Natural Number ◽  
Gap Series ◽  
Image Position

Theorem. Letbe a function holomorphic in the disk, wherep is a natural number andIfthen then f(z) assumes every complex value infinitely often in every sector.The purpose of this note is to prove the above result. To do this, we first observe that from the condition a<∞, we can easily show that the derivative f′(z) satisfying

On the Representation of Boolean Algebras

1962 ◽  
Vol 5 (1) ◽  
pp. 37-41 ◽  
Author(s):  
Günter Bruns
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Two Systems ◽  
Image Position

Let B be a Boolean algebra and let ℳ and n be two systems of subsets of B, both containing all finite subsets of B. Let us assume further that the join ∨M of every set M∊ℳ and the meet ∧N of every set N∊n exist. Several authors have treated the question under which conditions there exists an isomorphism φ between B and a field δ of sets, satisfying the conditions:

scholarly journals A class of harmonically convergent sets

1975 ◽  
Vol 20 (3) ◽  
pp. 301-304
Author(s):  
Torleiv Kløve
Keyword(s):  
Natural Numbers ◽  
Lower Case ◽  
Image Position

Following Craven (1965) we say that a set M of natural numbers is harmonically convergent if converges, and we call μ(M) the harmonic sum of M. (Craven defined these concepts for sequences rather than sets, but we found it convenient to work with sets.) Throughout this paper, lower case italics denote non-negative integers.

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