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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E1110.

E1110: density transfer and an incomplete claim summary

Bases p>q>=2 are coprime integers, not necessarily primes, with (3,2) excluded. Terms p^i q^j have nonnegative exponents and form a finite antichain under divisibility. Distinguish infinitely many exceptions coprime to pq from an infinite pairwise-coprime family; the latter implies the former after discarding finitely many terms. Neither replaces the density question.

Erdős--Lewin Theorem 1 has a short inspected proof. In a representation of ps, at most one term has p-exponent zero. Exactly one is impossible modulo p, since a q-power is a unit. Thus all terms are divisible by p, and dividing would represent s. So a non-representable s stays so after multiplication by p or q, also for composite coprime bases. This repeats prime factors and gives no pairwise-coprime family.

There is a complete additional implication: density zero of representables gives an infinite pairwise-coprime family of exceptions prime to pq. After choosing finitely many terms let M be pq times their product. The progression 1 modulo M has density 1/M and cannot eventually lie in a density-zero set. Choose a larger non-representable member and repeat. No independence assumption is used.

The Yu--Chen ranges reported in the held page leave five density pairs: (5,2),(7,2),(9,2),(4,3),(5,3). This enumeration is conditional on that range account. The publisher preview was reachable but omitted the mathematical hypotheses in its rendered text; it did not independently confirm the ranges. Its abstract only confirms partial answers. The full paper was not acquired. Positive lower density claimed for three of the pairs does not determine a natural density or prove that it exists. The three exceptional coprime claims (5,2),(9,2),(5,3), combined with the other ranges, would finish only the second question.

The unrestricted rectangle implication in the submission sketch on the claim page is false: with $p=5$, $q=2$, $a=M=1$, the singleton antichain $\{(0,0)\}$ has sum $1\equiv5\pmod2$ but not sum $5$. This is a counterexample only to the unrestricted sentence.

The claimed low-row interface includes $p^a<q^M$, which this example violates. The embedding uses both inequalities

$$p^a<q^M,\qquad cq^M<(p-1)p^a.$$

The second places all first coordinates at most $a$; the first permits the claimed low-row singleton conclusion. Removing $(a,0)$ and shifting the high rows down by $M$ then gives a representation of the seed $c$. For pairwise coprimality, the reported recursion also chooses $p^a\equiv q^M\equiv1\pmod D$, where $D$ is the product of prior terms, and uses that every prime factor of $c+1$ divides $pq$.

These are the reported source interfaces, not a new proof inspection. The primary low-row induction and the simultaneous power-window and period construction have not been audited here. The unrestricted counterexample gives no negative verdict on those hypotheses or on the underlying development, whose standing remains claimed.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/diophantine_problems/erdos_1996_d_complete_sequences_integers/_index|erdos 1996 d complete sequences integers]]: Section 2 Theorem 1 proof and Corollary; closing p.840 questions. Three-base proof material not used. (proof reading).