Erdős Problem #1200

Erdős Problem 1200 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

Does an absolute constant $C>0$ exist such that, for all sufficiently large $x$, one can select distinct primes $p_1<\cdots<p_k<x$ and residues $a_i\pmod{p_i}$ satisfying \[\sum_{i=1}^k\frac1{p_i}<C,\qquad [1,x)\cap\mathbb Z\subseteq\bigcup_{i=1}^k(a_i+p_i\mathbb Z)?\]

CONTEXT AND SCOPE

This replaces the earlier statement-only record with an elementary mathematical attempt. The arguments below establish only their stated partial results; no novelty or full solution is claimed.

The page says every integer n<x; the finite positive interval is the intended interpretation of this covering question.

ATTACK PLAN AND WORK

Let $M=\lceil x\rceil-1$, so the interval to be covered is $\{1,\ldots,M\}$. One class modulo $p_i$ covers at most $M/p_i+1$ of these integers. Therefore every proposed cover satisfies \[1\le\sum_{i=1}^k\frac1{p_i}+\frac kM.\tag{1}\] In particular a construction using $o(M)$ primes needs reciprocal mass at least $1-o(1)$. The usual fact $\pi(x)=o(x)$ gives this condition for the available primes as well; it is only a necessary lower threshold, not an impossibility result.

There is also an exact greedy estimate for any prescribed distinct primes. If $U$ integers remain uncovered, averaging over the $p$ residue classes finds one covering at least $U/p$ of them. Choosing such a class successively gives \[U_{\rm final}\le M\prod_i(1-1/p_i).\tag{2}\] No independence of the uncovered residue patterns is assumed in this averaging. However $2$ need not be below one when the reciprocal sum is bounded: for large selected primes the product is of constant scale, leaving an upper bound of order $M$. Failure of this bound to prove coverage is not a lower bound on what a better choice of residues can cover.

VERIFICATION AND REMAINING GAP

Exploit correlated choices of residues to cover the whole interval with bounded reciprocal mass. The necessary mass bound and elementary greedy upper bound on leftovers do not construct such a cover or disprove its existence.

SOURCES

https://www.erdosproblems.com/1200

FINAL

UNRESOLVED. The full source problem remains outside the scope of the proved partial results.

COMPLETION ESTIMATE

$6\%$. This is a conservative subjective estimate toward the whole problem, not a probability that the argument is correct or a claim that a fixed fraction of the problem has been solved.