New notes on the equation \(d(n)=d(\varphi(n))\) and the inequality \(d(n)>d(\varphi(n))\) (Q6634062)
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scientific article; zbMATH DE number 7939914
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | New notes on the equation \(d(n)=d(\varphi(n))\) and the inequality \(d(n)>d(\varphi(n))\) |
scientific article; zbMATH DE number 7939914 |
Statements
New notes on the equation \(d(n)=d(\varphi(n))\) and the inequality \(d(n)>d(\varphi(n))\) (English)
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6 November 2024
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Let \( n \) be a positive integer and, \( d(n) \) and \( \varphi (n) \) denote the number of positive divisors of \( n \) and the Euler's phi function of \( n \), respectively. In the paper under review, the authors use a clever combination of techniques in elementary number theory to prove various conditional and unconditional results about the solvability of the Diophantine equation \( d(n)=d(\varphi (n)) \) and the inequality \( d(n)< d(\varphi (n)) \). Furthermore, the authors present some of the interesting open problems in this research direction.
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divisor function
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Euler's function
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prime numbers
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Diophantine equations
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