gpt pro 5.4 attempt on Math-5893, version 3

Problem #783

1. ROUND-4 OBJECTIVE

I pursue path (C). I am not trying to re-solve the asymptotic value problem from scratch; Round 3 already identified the right corrected asymptotic statement and explained how Tao's repaired 2026 argument proves it. The genuine remaining issue is a status correction: Round 3 incorrectly labeled the investigation as fully complete for Problem #783. That overstates what has been achieved, because the original Erdős problem asks for the choice of the minimizing set $A$, whereas the argument supplied in Round 3 only determines the asymptotic minimum value \[ F_N(C):=\min_A |U_A(N)|. \]

So the Round-4 objective is to identify this fatal mismatch precisely, formulate the minimal corrected theorem that has actually been proved, and separate it rigorously from the still-open optimizer-classification problem.

2. Round-3 FOUNDATION USED

I use the following Round-3 conclusions as fixed input.

1. The explicit finite counterexample at $(N,C)=(30,1)$ disproving the literal exact prescription \[ \text{``take the consecutive largest primes''.} \]
2. The asymptotic extremal-value theorem \[ F_N(C)=\left(\rho(e^C)+o(1)\right)N, \qquad (N\to\infty), \] proved by combining Tao's 2026 lower bound for arbitrary pairwise coprime sets with the prime-tail upper bound.
3. The exact-budget prime-tail construction and the identification of its uncovered set with a smooth-number counting function.

I do not reprove any of these facts below.

3. NEW INSIGHT / TOOL (ROUND-4)

The new ingredient in this round is a precise logical separation lemma: knowledge of the asymptotic minimum value \[ F_N(C)=\left(\rho(e^C)+o(1)\right)N \] does not determine, even asymptotically, which admissible sets $A$ attain or nearly attain that minimum. In particular, it does not answer the original phrasing \[ \text{``What choice of such an $A$ minimises \dots ?''} \] except in the weakened sense "a prime tail is asymptotically value-optimal up to $o(N)$".

Thus the mathematically correct outcome is:

1. the exact finite optimizer prescription proposed by Erdős is false;
2. the asymptotic minimum value is known;
3. the original exact optimizer-classification problem remains open.

4. ATTACK PLAN (ROUND-4)

The exact gap left after Round 3 is not a missing estimate inside the asymptotic proof; it is a mismatch between:

1. the original target --- identify the minimizing set(s) $A$;
2. the proved statement --- identify only the asymptotic minimum value $F_N(C)$.

So I will prove the following three claims.

1. (i) Round 3 does not imply a full solution of the original problem.
2. (ii) The strongest theorem actually established so far is the asymptotic value theorem for $F_N(C)$.
3. (iii) The minimal corrected final status for Problem #783 is therefore UNRESOLVED (BUT STRICTLY ADVANCED).

This overcomes the main logical obstacle: it prevents us from conflating a full solution of a weaker asymptotic problem with a full solution of the original optimizer problem.

5. WORK (ROUND-4)

5.1 Restating the original target exactly.

Problem #783 asks:

Fix some constant $C>0$ and let $N$ be large. Let $A\subseteq\{2,\dots,N\}$ be such that $(a,b)=1$ for all distinct $a,b\in A$ and \[ \sum_{n\in A}\frac1n\le C. \] What choice of such an $A$ minimises the number of integers $m\le N$ not divisible by any $a\in A$?

This is an optimizer-classification question: it asks for the choice of $A$, not merely for the asymptotic size of the minimum.

5.2 What Round 3 actually proves.

Round 3 proves that if \[ F_N(C):=\min\left\{|U_A(N)|:\ A\subseteq\{2,\dots,N\}\ \text{pairwise coprime},\ \sum_{a\in A}\frac1a\le C\right\}, \] then \[ F_N(C)=\left(\rho(e^C)+o(1)\right)N. \] This determines the minimum value asymptotically.

It also gives an admissible family of prime tails $A_N$ with \[ |U_{A_N}(N)|=\left(\rho(e^C)+o(1)\right)N, \] so prime tails are asymptotically optimal in value.

But this does not show any of the following:

1. that for each large $N$ a prime tail is an exact minimizer;
2. that every minimizer is a prime tail;
3. that every near-minimizer must be close to a prime tail in any natural metric;
4. that one can classify the finite-$N$ minimizers.

Therefore Round 3 does not solve the original question as stated.

5.3 Formal separation of value and optimizer problems.

I now record the precise logical implication.

Lemma. Knowledge of the asymptotic minimum value \[ F_N(C)=\left(\rho(e^C)+o(1)\right)N \] does not by itself determine the minimizing sets for Problem #783.

Proof. The quantity $F_N(C)$ is a single number. The optimizer problem asks for the subset or family of subsets \[ \mathcal M_N(C):=\left\{A:\ |U_A(N)|=F_N(C)\right\}. \] Even exact knowledge of $F_N(C)$ leaves open the structure of $\mathcal M_N(C)$. In particular, two distinct admissible sets $A,B$ may satisfy \[ |U_A(N)|=|U_B(N)|=F_N(C), \] or one may have a unique minimizer, or many unrelated near-minimizers. None of this can be inferred from the scalar asymptotic formula for $F_N(C)$ alone.

Therefore the asymptotic value theorem does not answer the optimizer-classification question posed by Problem #783. $\square$

5.4 Explicit obstruction to exact prime-tail optimality.

Round 3 already gave the explicit counterexample at $(N,C)=(30,1)$ showing that the literal exact finite- $N$ prescription "take the consecutive largest primes" is false. This immediately rules out the strongest possible interpretation of Erdős's suggestion.

Hence the original problem cannot presently be declared solved by exhibiting that prescription.

5.5 Minimal corrected theorem.

Combining the Round-3 asymptotic theorem with the previous paragraph, the strongest mathematically justified corrected statement is the following.

Corrected theorem. For every fixed $C>0$, \[ F_N(C)=\left(\rho(e^C)+o(1)\right)N \qquad (N\to\infty). \] Moreover, a tail of large primes is asymptotically optimal in value, but the literal finite- $N$ prescription of consecutive largest primes is not always an exact minimizer.

This is a theorem about the asymptotic minimum value; it is not a classification of the minimizing sets.

5.6 Correct final status for the original problem.

Since the original problem asks for the choice of minimizing $A$, and since neither Round 3 nor the cited 2026 discussion classifies exact minimizers for all large $N$, the original problem remains open.

What has been achieved is a strict advance: the asymptotic minimum value is known, and one natural exact candidate family has been disproved as a universal optimizer.

Thus the proper final status is: \[ \boxed{\text{Problem \#783 remains open in its original optimizer form, but its asymptotic value problem is solved.}} \]

6. ADVERSARIAL VERIFICATION

I checked the possible failure points in this status correction.

1. Am I discarding Round-3 mathematics? No. I keep the asymptotic theorem intact. I only correct the over-strong interpretation of what that theorem resolves.
2. Could the asymptotic value theorem implicitly determine minimizers? No. A scalar minimum value does not determine the argmin set without an additional uniqueness or rigidity theorem, and no such theorem was proved in Round 3.
3. Does the explicit finite counterexample really matter? Yes. It rules out the strongest exact finite formulation and shows that even if prime tails are asymptotically optimal in value, the finite optimizer question is subtler.
4. Is this merely semantic rather than mathematical? No. The distinction between proving \[ F_N(C)=\left(\rho(e^C)+o(1)\right)N \] and classifying minimizers is a genuine logical distinction between two different mathematical statements.
5. Could one still call the original problem solved in a weak sense? Only after changing the problem statement to the asymptotic value version. For the original optimizer question, that would be inaccurate.

7. FINAL

UNRESOLVED (BUT STRICTLY ADVANCED).

Round 3 correctly established the asymptotic minimum-value theorem \[ F_N(C)=\left(\rho(e^C)+o(1)\right)N, \] and correctly disproved the literal exact finite- $N$ consecutive-largest-primes prescription. However, this does not fully answer the original Erdős question, which asks for the choice of minimizing admissible set $A$.

Therefore the correct Round-4 conclusion is:

1. the exact finite optimizer problem remains open;
2. the asymptotic minimum value is solved;
3. the proper corrected status of Problem #783 is open, with a solved asymptotic value variant.

8. COMPLETION ESTIMATE

COMPLETION: 85%

9. REFERENCES

1. [EP783 ] T. F. Bloom, Erdős Problem #783, https://www.erdosproblems.com/783, accessed 2026-03-09.
2. [Er73 ] P. Erdős, Problems and results on combinatorial number theory, in A Survey of Combinatorial Theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), North-Holland, Amsterdam, 1973, pp. 117--138.
3. [Tao26 ] T. Tao, Sieving by coprime numbers, repaired 2026 version discussed in the Problem #783 forum thread.
4. [Hi87 ] A. Hildebrand, Quantitative mean value theorems for nonnegative multiplicative functions II, Acta Arith. 48 $1987$, no. 3, 209--260.