Find out a process to generate pairs of distinct positive integer $m$, $n$ with $\phi(m) = \phi(n)$.
Attempt: The pairs $m=1, ~ n=2$; $m=3, ~n=4$ satisfy the problem. But I need a concrete solution of the problem to generate many pairs. Please help.
Find out a process to generate pairs of distinct positive integer $m$, $n$ with $\phi(m) = \phi(n)$.
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user1942348
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As phrased, you are done because you have found a pair, 2 in fact. However, if the question said "all", then... – Faraz Masroor Jun 28 '15 at 03:48
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@FarazMasroor Yes, I want to get a mathematical process, not the trial solutions. if I need "10 pairs" or "all", then problem arise. – user1942348 Jun 28 '15 at 03:53
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@Travis Travis and all I have edited my problem. – user1942348 Jun 28 '15 at 04:05
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If you want $10$ pairs, or infinitely many, note that if $k\ge 1$ then $\varphi(3\cdot 2^k)=\varphi(2^{k+1})$.
One can also generate infinitely many triples, playing a similar game with $5\cdot 2^k$, $3\cdot 2^{k+1}$, and $2^{k+2}$.
By playing with small numbers, we can find many other examples. For instance, $\varphi(7)=\varphi(9)$ and we can get an infinite family by decorating with powers of $2$. Or if powers of $2$ are uninteresting, we can use $7\cdot m$ and $9\cdot m$, where $m$ is divisible neither by $7$ nor by $3$.
To my knowledge there is no useful characterization of all pairs.
André Nicolas
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1just in case... odd primes $p$ and their values multiplied by two $2\cdot p$ have also always the same totient value? $\varphi(p)=\varphi(2p)$? – iadvd Jun 28 '15 at 04:12
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@iadvd: That is a nice example, since it gives an infinite family of infinite families, by multiplying $p$ and $2p$ by any odd $a$ relatively prime to $p$. And we do not need primes $p$, odd numbers $\gt 1$ will do. – André Nicolas Jun 28 '15 at 04:20
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Cool! I remembered that from previous questions I did here regarding the totient function, learning from you guys :) by the way, may I abuse from your time? I have a question regarding totient function with a bounty that will expire very soon, so it is a pity! I was looking for somebody able to give some hints about it... if you could someday have a look to it would be very appreciated! http://math.stackexchange.com/questions/1258928/eulers-totient-and-divisors-count-function-relationship-when-frac-varphin – iadvd Jun 28 '15 at 04:50