# Proof of the Infinitude of Odd Primitive Pseudoperfect Numbers

I'm interested in the infinitude of odd primitive pseudoperfect numbers. Richard K. Guy's book "Unsolved Problems in Number Theory 3rd edition" says that P. Erdős proved the infinitude of odd primitive pseudoperfect numbers.

Here, natural number $n$ called pseudoperfect number if one of distinct sums of some proper divisors of $n$ equals $n$. PseudoPerfect Number $n$ called primitive if any proper divisor of $n$ is not pseudoperfect.

Recently, I found a paper which perhaps prove this result. The paper is

S. J. Benkoski & P. Erdős, On weird and pseudoperfect numbers, _Math. Comput., vol. 28 (1974), 617-623. http://www.renyi.hu/~p_erdos/1974-24.pdf

In this paper, Theorem 3 proves the infinitude of odd primitive pseudoperfect numbers. But unfortunately, authors only outline the proof, and I can not understand their proof.

My questions are about their proof. I'm concentrated only on $B_k$.

1. How one can prove Lemma 1 in this paper? (I thought Vinogradov's three prime theorem and sieves can be used to prove this, but I failed.)

2. How one can prove Lemma 2 in this paper?

3. Why inequality (12) in this paper implies $B_k$ is pseudoperfect?

4. Are there any elementary proof of the infinitude of odd primitive pseudoperfect numbers? (Authors use the fact $p_{n+1} < (1+o(1))p_n$, and I can not prove this without PNT.)

5. I think the assertion which follows Lemma 2 "For $p_k\le x<3/2x<A_k$, the interval $(x,3/2x)$ always contains a divisor of $A_k$ and $B_k$." is false. Because in the case $x=A_k/p_k$, $x$ is the largest proper divisor of $A_k$. Is this assretion true?

I would appreciate it if you answer one or more above questions.

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Sorry, my question 3 was very easy if we proved Lemma 2. By the minimality of $f(k)$, we have $\sigma(B_k)/B_k<2+\epsilon$ with small $\epsilon>0$. By (12) and Lemma2, $2B_k$ is a distinct sum of divisors $B_k$. If this sum does not use a divisor $B_k$ itself, the sum is smaller than $\sigma(B_k)-B_k<(1+\epsilon)B_k$. This is a contradiction. – Yuta Suzuki Jul 17 '13 at 16:47
I (perhaps) succeeded in proving the infinitude of odd primitive pseudoperfect numbers. My proof largely follows Erdos-Benkoski's proof, but I modified Lemmas. In Lemma 1, I replaced the condition $m>cp_k$ with $m>cp_k\log p_k$. In Lemma 2, I replaced the condition $cp_k<m<\sigma(B_k)-cp_k$ with $cp_k\log p_k<m<B_k/p_k$. And $cp_k\log p_k<\sigma(B_k)-2B_k<B_k/p_k$ can be proven easily. – Yuta Suzuki Jul 18 '13 at 13:39