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Some unconventional problems in number theory

Some unconventional problems in number theory Acta Mathematica Academiae Scientia,um Hungaricae Tomus 33 (1--2), (1979), pp. 71--80. SOME UNCONVENTIONAL PROBLEMS IN NUMBER THEORY By P. ERDOS (Budapest), member of the Academy Dedicated to the 80th birthday of my friend George Alexits In the paper, we will mostly deal with arithmetic functions, primes, divisors, sieve processes and consecutive integers. 1. Letfbe an arithmetic function. The integer n is called a barrier forfif (1) m + f(m) < n for every m < n. Perhaps I should explain why I considered (1). In the early 1950's, van Wijn- gaarden told me the following conjecture. Put al(n) = a(n), the sum of divisors of n, and ak(n) = al(a~_i(n)). Is it true that there is essentially only one sequence a~(n) (k = 1, 2, 3 .... ) ? In other words, if m and n are distinct integers, are there integers k and l such that ak(m ) = al(n ) ? Such a conjecture is usually hopeless to prove Or disprove. Selfridge and others made some computer experiments and believe that the conjecture is false. I tried to find an airthmetic function for which an analogous conjecture is true and can be proved. Putf,(n) = n + http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

Some unconventional problems in number theory

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References (8)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01903382
Publisher site
See Article on Publisher Site

Abstract

Acta Mathematica Academiae Scientia,um Hungaricae Tomus 33 (1--2), (1979), pp. 71--80. SOME UNCONVENTIONAL PROBLEMS IN NUMBER THEORY By P. ERDOS (Budapest), member of the Academy Dedicated to the 80th birthday of my friend George Alexits In the paper, we will mostly deal with arithmetic functions, primes, divisors, sieve processes and consecutive integers. 1. Letfbe an arithmetic function. The integer n is called a barrier forfif (1) m + f(m) < n for every m < n. Perhaps I should explain why I considered (1). In the early 1950's, van Wijn- gaarden told me the following conjecture. Put al(n) = a(n), the sum of divisors of n, and ak(n) = al(a~_i(n)). Is it true that there is essentially only one sequence a~(n) (k = 1, 2, 3 .... ) ? In other words, if m and n are distinct integers, are there integers k and l such that ak(m ) = al(n ) ? Such a conjecture is usually hopeless to prove Or disprove. Selfridge and others made some computer experiments and believe that the conjecture is false. I tried to find an airthmetic function for which an analogous conjecture is true and can be proved. Putf,(n) = n +

Journal

Acta Mathematica Academiae Scientiarum HungaricaSpringer Journals

Published: Jun 21, 2005

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