Multiplicative independence

In number theory, two positive integers a and b are said to be multiplicatively independent[1] if their only common integer power is 1. That is, for integers n and m, $a^{n}=b^{m}$ implies $n=m=0$ . Two integers which are not multiplicatively independent are said to be multiplicatively dependent.

For example, 36 and 216 are multiplicatively dependent since $36^{3}=(6^{2})^{3}=(6^{3})^{2}=216^{2}$ and 6 and 12 are multiplicatively independent

Properties

Being multiplicatively independent admits some other characterizations. a and b are multiplicatively independent if and only if $\log(a)/\log(b)$ is irrational. This property holds independently of the base of the logarithm.

Let $a=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}$ and $b=q_{1}^{\beta _{1}}q_{2}^{\beta _{2}}\cdots q_{l}^{\beta _{l}}$ be the canonical representations of a and b. The integers a and b are multiplicatively dependent if and only if k = l, $p_{i}=q_{i}$ and ${\frac {\alpha _{i}}{\beta _{i}}}={\frac {\alpha _{j}}{\beta _{j}}}$ for all i and j.

Applications

Büchi arithmetic in base a and b define the same sets if and only if a and b are multiplicatively dependent.

Let a and b be multiplicatively dependent integers, that is, there exists n,m>1 such that $a^{n}=b^{m}$ . The integers c such that the length of its expansion in base a is at most m are exactly the integers such that the length of their expansion in base b is at most n. It implies that computing the base b expansion of a number, given its base a expansion, can be done by transforming consecutive sequences of m base a digits into consecutive sequence of n base b digits.

gollark: Anyway, to summarise the rest of it, as I'm sure you've seen, eventually the reshuffling thing began actually happening in earnest, and today another thing happened (I forgot what), various people left, more of them joined new esolangs without leaving, and palaiologos pushed ahead with stricter rule updates in a rather authoritarian way.

gollark: Yes.

gollark: Anyway, at some point palaiologos seems to have decided that the server needed a substantial overhaul.(For video purposes you should of course find screenshots of all this stuff; I think much of it was in staff chat though)

gollark: The only person to actually *leave* due to the initial thing was cyan, who remains on the other esolangs.

gollark: Many things happened, yes.

References

[2]

Bès, Alexis. "A survey of Arithmetical Definability". Archived from the original on 28 November 2012. Retrieved 27 June 2012.
Bruyère, Véronique; Hansel, Georges; Michaux, Christian; Villemaire, Roger (1994). "Logic and p-recognizable sets of integers" (PDF). Bull. Belg. Math. Soc. 1: 191--238.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Bès, Alexis. "A survey of Arithmetical Definability". Archived from the original on 28 November 2012. Retrieved 27 June 2012.

[2] Bruyère, Véronique; Hansel, Georges; Michaux, Christian; Villemaire, Roger (1994). "Logic and p-recognizable sets of integers" (PDF). Bull. Belg. Math. Soc. 1: 191--238.