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Let $\sigma(x)$ denote the sum of the divisors of $x$, and call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A number $N$ is called perfect if $\sigma(N)=2N$.

According to this paper, it was B. Hornfeck who proved that different odd perfect numbers, $n_1 = {{p_1}^{i_1}}{m_1}^2 \neq n_2 = {{p_2}^{i_2}}{m_2}^2$ have distinct $m_i$, where $p_1$ and $p_2$ are primes. That is:

Claim: If $n_1 = {{p_1}^{i_1}}{m_1}^2$ and $n_2 = {{p_2}^{i_2}}{m_2}^2$ are odd perfect numbers and $m_1 = m_2$, then $n_1 = n_2$.

Proof of Claim

Let $m_1 = m_2$. Since $n_1 = {{p_1}^{i_1}}{m_1}^2$ and $n_2 = {{p_2}^{i_2}}{m_2}^2$ are odd perfect numbers, we have

$$I(n_1) = I({p_1}^{i_1})I({m_1}^2) = 2$$ $$I(n_2) = I({p_2}^{i_2})I({m_2}^2) = 2$$

Therefore:

$$\frac{2}{I({p_1}^{i_1})} = I({m_1}^2) = I({m_2}^2) = \frac{2}{I({p_2}^{i_2})}.$$

Consequently:

$$I({p_1}^{i_1}) = I({p_2}^{i_2})$$

from which it follows that

$${p_1}^{i_1} = {p_2}^{i_2}$$

since powers of primes are solitary.

We conclude that $n_1 = n_2$.

QED.

My question is: Does anybody here have a reference to B. Hornfeck's paper, where this claim is proved?

Thanks!

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The paper cites the book Not Always Buried Deep which cites this paper:

Bernhard Hornfeck, Zur Dichte der Menge der vollkommenen Zahlen, Archiv der Mathematik October 1955, Volume 6, Issue 6, pp 442-443, DOI: 10.1007/BF01901120.

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  • $\begingroup$ Thanks for checking! An upvote for now. Will double-check your answer in a few. $\endgroup$ – Jose Arnaldo Bebita-Dris Jun 11 '16 at 13:50
  • $\begingroup$ I see no reference to a result by Hornfeck on page 251 of Not Always Buried Deep, as mentioned in that paper. $\endgroup$ – lhf Jun 11 '16 at 14:05

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