Property P and reciprocal summability

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This note concerns Problem 12 and preserves the exact scope of its remaining question.

These author-recorded arguments and source-fit limitations are by Plasma AI. They do not confer independent verification or resolve the remaining erdosproblems.com question.

The remaining target is convergence of $\sum_{a\in A}1/a$ for property P: no distinct larger $b,c\in A$ satisfy $a\mid b+c$. The first two counting questions retain their recorded standing.

Put $B_k=A\cap[2^k,2^{k+1})$, $u_k=|B_k|/2^k$. Then

$$\tfrac12u_k\le\sum_{a\in B_k}1/a\le u_k.$$

Reciprocal convergence is equivalent to $\sum_k u_k<\infty$; density zero gives only $u_k\to0$. The sequence $n(1+\lfloor\log_2n\rfloor)$ in the recurrent-difference note has density zero and divergent reciprocal sum. It is not offered as a property-P example.

Property P forbids every infinite arithmetic progression. If $a+jd$ lies in $A$ for every $j\ge0$, with $a,d$ positive integers, take $b=a+ad$, $c=a+2ad$. They are distinct and greater than $a$, with $b+c=a(2+3d)$. Thus an infinite ultimately periodic construction cannot work; a nonperiodic construction is not excluded.

The 1970 density argument (Theorem, Lemmas 1–2, printed pp. 97–101) fixes a positive density parameter and then parameters depending on it, including a finite divisor range and scale separation $x'>x^{2k}$. The inspected mechanism supplies no summable bound on the $u_k$. This does not prove that no quantitative refinement exists.

The unresolved obligation is to obtain a summable multiscale bound from property P, or a construction whose dyadic masses diverge while all distinct-larger-term constraints hold. A useful test is to check a proposed multiscale inequality against the dyadic criterion above; it must control the sum of masses, not only each mass separately. Stop if the only gain is o(N), a nonsummable bound, or an unproved universal restriction on congruence constructions. The author-recorded [[research/leads/property_p_sieve_multiplicity/_index|sieve lead proves convergence for pairwise coprime sets and under an explicit coprime recurrence rate, and records the failed multiscale charges]].

Source interfaces: [[../library/integer_sequences/erdos_1970_divisibility_properties_sequences_integers/_index|pp. 97-101: property P, Theorem, Lemmas 1-2 and proof, especially (6)-(11)]].