On primes and practical numbers
| Autor: | Andreas Weingartner, Carl Pomerance |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Practical number
Algebra and Number Theory Conjecture Mathematics - Number Theory Distribution (number theory) 010102 general mathematics 0102 computer and information sciences 01 natural sciences Upper and lower bounds Prime (order theory) Combinatorics Number theory Integer 010201 computation theory & mathematics FOS: Mathematics Subset sum problem Number Theory (math.NT) 0101 mathematics Mathematics 11N25 (11N37) |
| DOI: | 10.48550/arxiv.2007.11062 |
| Popis: | A number $n$ is practical if every integer in $[1,n]$ can be expressed as a subset sum of the positive divisors of $n$. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and Weingartner. In addition, essentially proving a conjecture of Margenstern, we show that all large odd numbers are the sum of a prime and a practical number. We also consider an analogue of the prime $k$-tuples conjecture for practical numbers, proving the "correct" upper bound, and for pairs, improving on a lower bound of Melfi. |
| Databáze: | OpenAIRE |
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