MHB Prove that n is not divisible by 105.

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The discussion centers on proving that an odd perfect number, denoted as n, cannot be divisible by 105. The number 105 factors into 3, 5, and 7, and the argument suggests examining the properties of divisors and their sums in relation to these prime factors. It highlights that if n were divisible by 105, it would contradict the conditions necessary for n to be an odd perfect number. Additionally, the existence of odd perfect numbers remains an open question in number theory. The conclusion emphasizes the need for further exploration into the properties of odd perfect numbers and their divisibility.
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Problem: Suppose that a natural number $n$ is an odd perfect number, i.e.:

$n$ is odd and $n$ is equal to the sum of all its positive divisors (including $1$ and excluding $n$).Prove that $n$ is not divisible by $105$.P.S.: To this day, no one knows, whether such a number exists. Here
is a comment on the subject from Wolfram Mathworld.
 
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Suggested solution:
Suppose that $n$ is divisible by $105 = 3 \cdot 5 \cdot 7$. Consider the prime factorization of $n$:

\[n = 3^{\alpha_1}\cdot 5^{\alpha_2 } \cdot 7^{\alpha_3}\cdot p_4^{\alpha_4}\cdot ...\cdot p_k^{\alpha_k},\: \: \: \alpha_1,\alpha_2 , \alpha_3 \geq 1.\]

Let $S(n)$ be the sum of all positive divisors of $n$ (including $1$ and $n$):

\[S(n)= n\left ( 1+\frac{1}{3}+...+\frac{1}{3^{\alpha_1}} \right )\left ( 1+\frac{1}{5}+...+\frac{1}{5^{\alpha_2}} \right )\left ( 1+\frac{1}{7}+...+\frac{1}{7^{\alpha_3}} \right )\left ( 1+\frac{1}{p_4}+...+\frac{1}{p_4^{\alpha_4}} \right )...\left ( 1+\frac{1}{p_k}+...+\frac{1}{p_k^{\alpha_k}} \right )\]

Since $n$ is an odd perfect number $S(n) = 2n$ and $S(n)$ is not divisible by $4$. Since all primes in the decomposition of $n$ are odd, $\alpha_1$ and $\alpha_3$ are at least $2$, - otherwise

$\left ( 1+\frac{1}{3}+...+\frac{1}{3^{\alpha_1}} \right )=\frac{4}{3}$ and $\left ( 1+\frac{1}{7}+...+\frac{1}{7^{\alpha_3}} \right )=\frac{8}{7}$ and $S(n)$ is divisible by $4$. Finally:
\[2 = \frac{S(n)}{n}= \left ( 1+\frac{1}{3}+\frac{1}{3^{2}} \right )\left ( 1+\frac{1}{5}\right )\left ( 1+\frac{1}{7}+\frac{1}{7^2} \right )=\frac{13}{9}\cdot \frac{6}{5} \cdot \frac{57}{49}= \frac{4446}{2205}>2.\]
Contradiction.
 
lfdahl said:
Suggested solution:
Suppose that $n$ is divisible by $105 = 3 \cdot 5 \cdot 7$. Consider the prime factorization of $n$:

\[n = 3^{\alpha_1}\cdot 5^{\alpha_2 } \cdot 7^{\alpha_3}\cdot p_4^{\alpha_4}\cdot ...\cdot p_k^{\alpha_k},\: \: \: \alpha_1,\alpha_2 , \alpha_3 \geq 1.\]

Let $S(n)$ be the sum of all positive divisors of $n$ (including $1$ and $n$):

\[S(n)= n\left ( 1+\frac{1}{3}+...+\frac{1}{3^{\alpha_1}} \right )\left ( 1+\frac{1}{5}+...+\frac{1}{5^{\alpha_2}} \right )\left ( 1+\frac{1}{7}+...+\frac{1}{7^{\alpha_3}} \right )\left ( 1+\frac{1}{p_4}+...+\frac{1}{p_4^{\alpha_4}} \right )...\left ( 1+\frac{1}{p_k}+...+\frac{1}{p_k^{\alpha_k}} \right )\]

Since $n$ is an odd perfect number $S(n) = 2n$ and $S(n)$ is not divisible by $4$. Since all primes in the decomposition of $n$ are odd, $\alpha_1$ and $\alpha_3$ are at least $2$, - otherwise

$\left ( 1+\frac{1}{3}+...+\frac{1}{3^{\alpha_1}} \right )=\frac{4}{3}$ and $\left ( 1+\frac{1}{7}+...+\frac{1}{7^{\alpha_3}} \right )=\frac{8}{7}$ and $S(n)$ is divisible by $4$. Finally:
\[2 = \frac{S(n)}{n}= \left ( 1+\frac{1}{3}+\frac{1}{3^{2}} \right )\left ( 1+\frac{1}{5}\right )\left ( 1+\frac{1}{7}+\frac{1}{7^2} \right )=\frac{13}{9}\cdot \frac{6}{5} \cdot \frac{57}{49}= \frac{4446}{2205}>2.\]
Contradiction.
why $S(n)=2n\, ?$ ,can you give me an example ?
 
Albert said:
why $S(n)=2n\, ?$ ,can you give me an example ?

Yes: Take $n = 6$, which is an even perfect number, because the sum of its divisors (except for $n$) is:

$1+2+3 = 6$. The proof defines $S(n)$ as the sum of all divisors of $n$ including $n$ itself.

Thus: $S(6) = 1+2+3+6 = 12 = 2 \cdot 6$.

I would like to give you an example with $n$ odd, but this would be a hard task, since math researchers have shown, that the
smallest possible odd perfect number $n > 10^{1500}$.
 
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