$n \phi (m) = m\phi (n)$ implies $n=m$ (Euler totient)

Find all positive integers such that $n\phi(m)=m\phi(n)$ where $\phi$ is the Euler totient function.

I think that $n\phi(m)=m\phi(n)$ implies $n=m$ and that I should use prime factorization of n and m to show the $n=m$. So after writing n and m in their prime factorization I have: $n\phi(m)=m\phi(n) \Longleftrightarrow nm(\prod \limits_{i=1}^{k}(1-1/p_i)=mn(\prod\limits_{i=1}^{l}(1-1/q_i) \Longleftrightarrow\prod \limits_{i=1}^{k}(1-1/p_i) =\prod\limits_{i=1}^{l}(1-1/q_i)$

• Note that this is equivalent to $\phi(n)/n = \phi(m)/m$. Look at the values of $\phi(n)/n$ for some range of $n$. It shouldn't take you very long to notice that there are repeated values.... – mjqxxxx May 6 '18 at 17:55
• More specifically, the fact that $\phi(n)/n = \prod_{p | n} (1-1/p)$ makes it obvious that the value only depends on which primes divide $n$... not their multiplicities. – mjqxxxx May 6 '18 at 17:58

Consider the equality $$\prod \limits_{i=1}^{k}\frac{p_i-1}{p_i} =\prod\limits_{i=1}^{l}\frac{q_i-1}{q_i}$$ with $p_1<\ldots <p_k$. Since $p_k$ is in the denominator of the LHS, it must equal to one of the $q_i$. Divide on both sides by $\frac{p_k-1}{p_k}$ and repeat. We get that $k=l$ and $p_i$, $q_i$ are equal in some order. Therefore $\frac{\phi(m)}{m}=\frac{\phi(n)}{n}$ if and only if $m$, $n$ have the same factors

• thank you so much! – tommy_m May 6 '18 at 18:52

Your claim is false, but I can't quite tell if my modified version of it is always true:

$$m\phi(n) = n\phi(m) \iff {\rm rad}(m) = {\rm rad}(n)$$

where ${\rm rad}(k)$ is the radical of the integer, the square-free number that is divisible by every prime that divides $k$.

As a simple counterexample to the original claim, take $n=3, m=9$.

Sun May  6 12:29:02 PDT 2018
m : 2 Phi(m) 1           n : 4 Phi(n) 2
m : 2 Phi(m) 1           n : 8 Phi(n) 4
m : 4 Phi(m) 2           n : 8 Phi(n) 4
m : 3 Phi(m) 2           n : 9 Phi(n) 6
m : 6 Phi(m) 2           n : 12 Phi(n) 4
m : 2 Phi(m) 1           n : 16 Phi(n) 8
m : 4 Phi(m) 2           n : 16 Phi(n) 8
m : 8 Phi(m) 4           n : 16 Phi(n) 8
m : 6 Phi(m) 2           n : 18 Phi(n) 6
m : 12 Phi(m) 4           n : 18 Phi(n) 6
m : 10 Phi(m) 4           n : 20 Phi(n) 8
m : 6 Phi(m) 2           n : 24 Phi(n) 8
m : 12 Phi(m) 4           n : 24 Phi(n) 8
m : 18 Phi(m) 6           n : 24 Phi(n) 8
m : 5 Phi(m) 4           n : 25 Phi(n) 20
m : 3 Phi(m) 2           n : 27 Phi(n) 18
m : 9 Phi(m) 6           n : 27 Phi(n) 18
m : 14 Phi(m) 6           n : 28 Phi(n) 12
m : 2 Phi(m) 1           n : 32 Phi(n) 16
m : 4 Phi(m) 2           n : 32 Phi(n) 16
m : 8 Phi(m) 4           n : 32 Phi(n) 16
m : 16 Phi(m) 8           n : 32 Phi(n) 16
m : 6 Phi(m) 2           n : 36 Phi(n) 12
m : 12 Phi(m) 4           n : 36 Phi(n) 12
m : 18 Phi(m) 6           n : 36 Phi(n) 12
m : 24 Phi(m) 8           n : 36 Phi(n) 12
m : 10 Phi(m) 4           n : 40 Phi(n) 16
m : 20 Phi(m) 8           n : 40 Phi(n) 16
m : 22 Phi(m) 10           n : 44 Phi(n) 20
m : 15 Phi(m) 8           n : 45 Phi(n) 24
m : 6 Phi(m) 2           n : 48 Phi(n) 16
m : 12 Phi(m) 4           n : 48 Phi(n) 16
m : 18 Phi(m) 6           n : 48 Phi(n) 16
m : 24 Phi(m) 8           n : 48 Phi(n) 16
m : 36 Phi(m) 12           n : 48 Phi(n) 16
m : 7 Phi(m) 6           n : 49 Phi(n) 42
m : 10 Phi(m) 4           n : 50 Phi(n) 20
m : 20 Phi(m) 8           n : 50 Phi(n) 20
m : 40 Phi(m) 16           n : 50 Phi(n) 20
m : 26 Phi(m) 12           n : 52 Phi(n) 24
m : 6 Phi(m) 2           n : 54 Phi(n) 18
m : 12 Phi(m) 4           n : 54 Phi(n) 18
m : 18 Phi(m) 6           n : 54 Phi(n) 18
m : 24 Phi(m) 8           n : 54 Phi(n) 18
m : 36 Phi(m) 12           n : 54 Phi(n) 18
m : 48 Phi(m) 16           n : 54 Phi(n) 18
m : 14 Phi(m) 6           n : 56 Phi(n) 24
m : 28 Phi(m) 12           n : 56 Phi(n) 24
m : 30 Phi(m) 8           n : 60 Phi(n) 16
m : 21 Phi(m) 12           n : 63 Phi(n) 36
m : 2 Phi(m) 1           n : 64 Phi(n) 32
m : 4 Phi(m) 2           n : 64 Phi(n) 32
m : 8 Phi(m) 4           n : 64 Phi(n) 32
m : 16 Phi(m) 8           n : 64 Phi(n) 32
m : 32 Phi(m) 16           n : 64 Phi(n) 32
m : 34 Phi(m) 16           n : 68 Phi(n) 32
m : 6 Phi(m) 2           n : 72 Phi(n) 24
m : 12 Phi(m) 4           n : 72 Phi(n) 24
m : 18 Phi(m) 6           n : 72 Phi(n) 24
m : 24 Phi(m) 8           n : 72 Phi(n) 24
m : 36 Phi(m) 12           n : 72 Phi(n) 24
m : 48 Phi(m) 16           n : 72 Phi(n) 24
m : 54 Phi(m) 18           n : 72 Phi(n) 24
m : 15 Phi(m) 8           n : 75 Phi(n) 40
m : 45 Phi(m) 24           n : 75 Phi(n) 40
m : 38 Phi(m) 18           n : 76 Phi(n) 36
m : 10 Phi(m) 4           n : 80 Phi(n) 32
m : 20 Phi(m) 8           n : 80 Phi(n) 32
m : 40 Phi(m) 16           n : 80 Phi(n) 32
m : 50 Phi(m) 20           n : 80 Phi(n) 32
m : 3 Phi(m) 2           n : 81 Phi(n) 54
m : 9 Phi(m) 6           n : 81 Phi(n) 54
m : 27 Phi(m) 18           n : 81 Phi(n) 54
m : 42 Phi(m) 12           n : 84 Phi(n) 24
m : 22 Phi(m) 10           n : 88 Phi(n) 40
m : 44 Phi(m) 20           n : 88 Phi(n) 40
m : 30 Phi(m) 8           n : 90 Phi(n) 24
m : 60 Phi(m) 16           n : 90 Phi(n) 24
m : 46 Phi(m) 22           n : 92 Phi(n) 44
m : 6 Phi(m) 2           n : 96 Phi(n) 32
m : 12 Phi(m) 4           n : 96 Phi(n) 32
m : 18 Phi(m) 6           n : 96 Phi(n) 32
m : 24 Phi(m) 8           n : 96 Phi(n) 32
m : 36 Phi(m) 12           n : 96 Phi(n) 32
m : 48 Phi(m) 16           n : 96 Phi(n) 32
m : 54 Phi(m) 18           n : 96 Phi(n) 32
m : 72 Phi(m) 24           n : 96 Phi(n) 32
m : 14 Phi(m) 6           n : 98 Phi(n) 42
m : 28 Phi(m) 12           n : 98 Phi(n) 42
m : 56 Phi(m) 24           n : 98 Phi(n) 42
m : 33 Phi(m) 20           n : 99 Phi(n) 60
m : 10 Phi(m) 4           n : 100 Phi(n) 40
m : 20 Phi(m) 8           n : 100 Phi(n) 40
m : 40 Phi(m) 16           n : 100 Phi(n) 40
m : 50 Phi(m) 20           n : 100 Phi(n) 40
m : 80 Phi(m) 32           n : 100 Phi(n) 40
Sun May  6 12:29:02 PDT 2018


=============================================================================

Sun May  6 12:23:56 PDT 2018
m : 2 =  2           n : 4 =  2^2
m : 2 =  2           n : 8 =  2^3
m : 4 =  2^2           n : 8 =  2^3
m : 3 =  3           n : 9 =  3^2
m : 6 =  2 3           n : 12 =  2^2 3
m : 2 =  2           n : 16 =  2^4
m : 4 =  2^2           n : 16 =  2^4
m : 8 =  2^3           n : 16 =  2^4
m : 6 =  2 3           n : 18 =  2 3^2
m : 12 =  2^2 3           n : 18 =  2 3^2
m : 10 =  2 5           n : 20 =  2^2 5
m : 6 =  2 3           n : 24 =  2^3 3
m : 12 =  2^2 3           n : 24 =  2^3 3
m : 18 =  2 3^2           n : 24 =  2^3 3
m : 5 =  5           n : 25 =  5^2
m : 3 =  3           n : 27 =  3^3
m : 9 =  3^2           n : 27 =  3^3
m : 14 =  2 7           n : 28 =  2^2 7
m : 2 =  2           n : 32 =  2^5
m : 4 =  2^2           n : 32 =  2^5
m : 8 =  2^3           n : 32 =  2^5
m : 16 =  2^4           n : 32 =  2^5
m : 6 =  2 3           n : 36 =  2^2 3^2
m : 12 =  2^2 3           n : 36 =  2^2 3^2
m : 18 =  2 3^2           n : 36 =  2^2 3^2
m : 24 =  2^3 3           n : 36 =  2^2 3^2
m : 10 =  2 5           n : 40 =  2^3 5
m : 20 =  2^2 5           n : 40 =  2^3 5
m : 22 =  2 11           n : 44 =  2^2 11
m : 15 =  3 5           n : 45 =  3^2 5
m : 6 =  2 3           n : 48 =  2^4 3
m : 12 =  2^2 3           n : 48 =  2^4 3
m : 18 =  2 3^2           n : 48 =  2^4 3
m : 24 =  2^3 3           n : 48 =  2^4 3
m : 36 =  2^2 3^2           n : 48 =  2^4 3
m : 7 =  7           n : 49 =  7^2
m : 10 =  2 5           n : 50 =  2 5^2
m : 20 =  2^2 5           n : 50 =  2 5^2
m : 40 =  2^3 5           n : 50 =  2 5^2
m : 26 =  2 13           n : 52 =  2^2 13
m : 6 =  2 3           n : 54 =  2 3^3
m : 12 =  2^2 3           n : 54 =  2 3^3
m : 18 =  2 3^2           n : 54 =  2 3^3
m : 24 =  2^3 3           n : 54 =  2 3^3
m : 36 =  2^2 3^2           n : 54 =  2 3^3
m : 48 =  2^4 3           n : 54 =  2 3^3
m : 14 =  2 7           n : 56 =  2^3 7
m : 28 =  2^2 7           n : 56 =  2^3 7
m : 30 =  2 3 5           n : 60 =  2^2 3 5
m : 21 =  3 7           n : 63 =  3^2 7
m : 2 =  2           n : 64 =  2^6
m : 4 =  2^2           n : 64 =  2^6
m : 8 =  2^3           n : 64 =  2^6
m : 16 =  2^4           n : 64 =  2^6
m : 32 =  2^5           n : 64 =  2^6
m : 34 =  2 17           n : 68 =  2^2 17
m : 6 =  2 3           n : 72 =  2^3 3^2
m : 12 =  2^2 3           n : 72 =  2^3 3^2
m : 18 =  2 3^2           n : 72 =  2^3 3^2
m : 24 =  2^3 3           n : 72 =  2^3 3^2
m : 36 =  2^2 3^2           n : 72 =  2^3 3^2
m : 48 =  2^4 3           n : 72 =  2^3 3^2
m : 54 =  2 3^3           n : 72 =  2^3 3^2
m : 15 =  3 5           n : 75 =  3 5^2
m : 45 =  3^2 5           n : 75 =  3 5^2
m : 38 =  2 19           n : 76 =  2^2 19
m : 10 =  2 5           n : 80 =  2^4 5
m : 20 =  2^2 5           n : 80 =  2^4 5
m : 40 =  2^3 5           n : 80 =  2^4 5
m : 50 =  2 5^2           n : 80 =  2^4 5
m : 3 =  3           n : 81 =  3^4
m : 9 =  3^2           n : 81 =  3^4
m : 27 =  3^3           n : 81 =  3^4
m : 42 =  2 3 7           n : 84 =  2^2 3 7
m : 22 =  2 11           n : 88 =  2^3 11
m : 44 =  2^2 11           n : 88 =  2^3 11
m : 30 =  2 3 5           n : 90 =  2 3^2 5
m : 60 =  2^2 3 5           n : 90 =  2 3^2 5
m : 46 =  2 23           n : 92 =  2^2 23
m : 6 =  2 3           n : 96 =  2^5 3
m : 12 =  2^2 3           n : 96 =  2^5 3
m : 18 =  2 3^2           n : 96 =  2^5 3
m : 24 =  2^3 3           n : 96 =  2^5 3
m : 36 =  2^2 3^2           n : 96 =  2^5 3
m : 48 =  2^4 3           n : 96 =  2^5 3
m : 54 =  2 3^3           n : 96 =  2^5 3
m : 72 =  2^3 3^2           n : 96 =  2^5 3
m : 14 =  2 7           n : 98 =  2 7^2
m : 28 =  2^2 7           n : 98 =  2 7^2
m : 56 =  2^3 7           n : 98 =  2 7^2
m : 33 =  3 11           n : 99 =  3^2 11
m : 10 =  2 5           n : 100 =  2^2 5^2
m : 20 =  2^2 5           n : 100 =  2^2 5^2
m : 40 =  2^3 5           n : 100 =  2^2 5^2
m : 50 =  2 5^2           n : 100 =  2^2 5^2
m : 80 =  2^4 5           n : 100 =  2^2 5^2
Sun May  6 12:23:56 PDT 2018