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Importance: Medium ✭✭

Author(s):

Subject:	Number Theory
	» Analytic Number Theory

Keywords:	Divisibility
	mimic function
	mimic number

Recomm. for undergrads: yes

Prize: Not yet declared

Posted	by:	facility_cttb@i...
	on:	June 20th, 2009

Problem

Let the notation $a|b$ denote '' $a$ divides $b$ ''. The mimic function in number theory is defined as follows [1].

Definition For any positive integer $\mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i}$ divisible by $\mathcal{D}$ , the mimic function, $f(\mathcal{D} | \mathcal{N})$ , is given by,

$f(\mathcal{D} | \mathcal{N}) = \sum_{i=0}^{n}\mathcal{X}_{i}(\mathcal{M}-\mathcal{D})^{i}$

By using this definition of mimic function, the mimic number of any non-prime integer is defined as follows [1].

Definition The number $m$ is defined to be the mimic number of any positive integer $\mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i}$ , with respect to $\mathcal{D}$ , for the minimum value of which $f^{m}(\mathcal{D} | \mathcal{N}) = \mathcal{D}$ .

Given these two definitions and a positive integer $\mathcal{D}$ , find the distribution of mimic numbers of those numbers divisible by $\mathcal{D}$ .

Again, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer $\mathcal{D}$ .

Bibliography

*[1] Malay Bhattacharyya, Sanghamitra Bandyopadhyay and U Maulik, Non-primes are recursively divisible, Acta Universitatis Apulensis 19 (2009).

* indicates original appearance(s) of problem.

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