No. The three conditions
$f:\mathbb N \mapsto \mathbb N, f(mn)=f(m)f(n)$
$f$ strictly increasing.
$f(2)=2$ imply the unique solution $f(n)=n$.
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- Mon Sep 22, 2014 8:13 am
- Forum: Higher Secondary Level
- Topic: Functional Equation (Canada 1969)
- Replies: 10
- Views: 8473
- Sun Sep 21, 2014 10:15 pm
- Forum: Higher Secondary Level
- Topic: Functional Equation (Canada 1969)
- Replies: 10
- Views: 8473
Re: Functional Equation (Canada 1969)
And I meant the part "$f:\mathbb N \mapsto \mathbb N, f(mn)=f(m)f(n) \Rightarrow$ Cauchy" is wrong, and provided a counter example. Even if you add the strictly increasing part, you'd have to prove it differently (like you did in the edit).
- Sun Sep 21, 2014 2:26 pm
- Forum: Higher Secondary Level
- Topic: Functional Equation (Canada 1969)
- Replies: 10
- Views: 8473
Re: Functional Equation (Canada 1969)
For example, $F(2)=3, F(3)=2$, and $F(p)=p$ for all other prime $p$ yields a solution.
- Sat Sep 20, 2014 11:08 pm
- Forum: Higher Secondary Level
- Topic: Functional Equation (Canada 1969)
- Replies: 10
- Views: 8473
Re: Functional Equation (Canada 1969)
The given is Cauchy's Functional Equation. Hence \(f(x)=x^n\) for any fixed \(n\in\mathbb{N}\). Now \(f(2)=2^n=2\) gives \(n=1\) so \(f(n)=n\). Wrong. For example the following set of functions defines a solution to $f(mn) = f(m)f(n), f: \mathbb N \mapsto \mathbb N$. \[f(\prod p_i ^{e_i}) = \prod f...
- Sat Sep 20, 2014 9:41 pm
- Forum: Higher Secondary Level
- Topic: Functional Equation (Canada 1969)
- Replies: 10
- Views: 8473
Re: Functional Equation (Canada 1969)
Hint:
The set $S_n = \{n^q | q \in \mathbb Q\}$ is dense $\forall n \in \mathbb N, n > 1$.
The set $S_n = \{n^q | q \in \mathbb Q\}$ is dense $\forall n \in \mathbb N, n > 1$.
- Sat Sep 20, 2014 8:03 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Shortlist-2011 N3
- Replies: 2
- Views: 3049
Re: IMO Shortlist-2011 N3
Another member of the "Great Set of Number Theoretical Functions."
Next step:
Next step:
- Fri Sep 19, 2014 3:12 pm
- Forum: Number Theory
- Topic: An awesome note by Masum bhai
- Replies: 11
- Views: 7981
Re: An awesome note by Masum bhai
Downloading this requires membership in a specific group. Post the direct link or upload it here.mutasimmim wrote:আমারও একখান আছে
http://www.facebook.com/groups/bmoandim ... 174446904/
- Sat Sep 13, 2014 11:11 pm
- Forum: Social Lounge
- Topic: একটি জিজ্ঞাসা এবং একটি অনুরধ
- Replies: 4
- Views: 4974
Re: একটি জিজ্ঞাসা এবং একটি অনুরধ
Well, you can already discuss them here (as long as there isn't a separate philosophy subforum). As the description says, social lounge is the place for discussing random topics.
- Tue Sep 09, 2014 11:28 pm
- Forum: International Mathematical Olympiad (IMO)
- Topic: IMO Marathon
- Replies: 184
- Views: 116877
Re: IMO Marathon
Or the square can be $100 \times 100$.
- Mon Sep 08, 2014 7:46 pm
- Forum: Secondary Level
- Topic: X's and 3
- Replies: 8
- Views: 6149
Re: X's and 3
If x is even, then after 3 steps it would go imaginary. What? If x is odd, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative. What? If x is a fraction, then $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ would be negative. What? If $x>1$, then, $x^{x^{x^{x^{\cdots x^{x^{3}}}}}}$ is a stictly increasing...