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I did the following proof and I was wondering if its valid. It feels wrong because I didn't actually test the case when purportedly $n$ is not prime, but please feel free to correct me.

Assume there exists $n$ such that $\phi(n) = 14$. Assume $n$ is prime. Then $\phi(n) = n-1$. Then here, n-1 = 14, so n = 15. We know since Euler's totient function is multiplicative that $\phi(xy)$ = $\phi(x)\phi(y)$, so $\phi(15) = \phi(3)\phi(5)$, but alas $\phi(3)\phi(5) = 2\cdot4 = 8 \ne 14$. If $n$ is not prime a similar argument follows since we know then that $n$ must be composed of prime numbers by the prime factorization theorem.

Arthur Collé
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    I don't see at all what "similar argument" you're implying for the case when $n$ isn't prime, could you explain? – Jack M Apr 14 '14 at 23:05
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    (Also, your argument for the prime case is a bit complicated. Assume $n$ is prime, then $n=15$, contradiction!). – Jack M Apr 14 '14 at 23:09

3 Answers3

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If a prime $p\mid n$, then $p-1\mid\phi(n)$. If $\phi(n)=14$, then, since the divisors of $14$ are $\{1,2,7,14\}$, $p\in\{2,3\}$. Thus, $n=2^a3^b$ and $\phi(n)$ is of the form $2^j3^k$. However, there is no factor of $7$ in $2^j3^k$.

robjohn
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    Where is the 3 coming from? – Arthur Collé Apr 14 '14 at 23:12
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    @ArthurCollé: the set of divisors of $\phi(n)$ is ${1,2,7,14}$, therefore, the possible set of primes that divide $n$ is ${2,3,8,15}$, which can be reduced to ${2,3}$ since neither $8$ nor $15$ are primes. – robjohn Apr 14 '14 at 23:16
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    Thanks. I see how $n = 2^a 3^b$ but then why is $\phi(n) = 2^a 3^{b-1}$? What equality establishes that relationship? – Arthur Collé Apr 14 '14 at 23:20
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    @ArthurCollé: if $n=\prod\limits_ip_i^{e_i}$, where $p_i$ is prime and $e_i\gt0$, then $\phi(n)=\prod\limits_i(p_i-1)p_i^{e_i-1}$. – robjohn Apr 14 '14 at 23:25
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We answer the question, but in a way that is far too convoluted for the particular numerical example of the post. We prove the following result.

Lemma: If $\varphi(n)=2q$, where $q$ is an odd prime, then $2q+1$ is prime.

An immediate consequence is that we cannot have $\varphi(n)=14$. For then $q=7$, but $2q+1$ is not prime.

We now prove the lemma. Suppose that $\varphi(n)=2q$. We divide the analysis into cases.

Case $1$: Maybe $n=2^km$, where $k\ge 3$ and $m$ is odd. Then by the multiplicativity oof $\varphi$, we have $\varphi(n)=\varphi(2^k)\varphi(m)$. This is impossible, since $\varphi(2^k)=2^{k-1}$, which is divisible by $4$. But $2q$ is not divisible by $4$.

Case $2$: Maybe $n=4m$, where $m$ is odd. Then $\varphi(n)=2\varphi(m)$. Since $2q$ is twice an odd number, it follows that $\varphi(m)$ is odd. This is only possible for an odd number $m$ if $m=1$. But then $\varphi(m)=2\ne 2q$.

Case $3$: Maybe $n=2m$ or $n=m$, where $m$ is an odd number. In either case, $\varphi(n)=\varphi(m)$.

Suppose that $m$ can be factored as a product $st$, where $s$ and $t$ are relatively prime, and neither $s$ nor $t$ is equal to $1$. Then $\varphi(s)$ and $\varphi(t)$ are each even, so $\varphi(m)$ is divisible by $4$. But $2q$ isn't, so we cannot have $\varphi(m)=2q$.

It remains to deal with the case where $m$ is a prime power, perhaps equal to $1$. If $m=1$, then $2q=2$, not twice an odd prime.

Now suppose that $m=p^k$, where $p$ is an odd prime, and $k\ge 1$. Then $\varphi(m)=p^{k-1}(p-1)$. This can be twice an odd prime in $2$ ways: (i) $k=1$ and $p-1$ is twice an odd prime, or (ii) $p=3$ and $k=2$.

In Case (i), $2q=p-1$, so $2q+1$ is prime. In Case (ii), $2q=6$, so again $2q+1$ is prime.

André Nicolas
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This does not hold. What kind of similar argument are you then talking about?

You can use the fact that $\phi$ is multiplicative. Assume $n = p_1^{a_1}p_2^{a_2}...p_t^{a_t}$, then $\phi(p_1^{a_1}p_2^{a_2}...p_t^{a_t})$ $\phi(p_1^{a_1})\phi(p_2^{a_2})...\phi(p_t^{a_t}) = 2 \cdot 7 = 1 \cdot 14$

Use this to arrive at a contradiction.

Improve
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