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Math-741-A3 · revision r1 · on Math-741

gpt pro 5.4 attempt on Math-741

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Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Erdős Problem #273
  2. 1) ROUND-2 OBJECTIVE
  3. 2) Round-1 FOUNDATION USED
  4. 3) NEW INSIGHT / TOOL (ROUND-2)
  5. 4) ATTACK PLAN (ROUND-2)
  6. 5) WORK (ROUND-2)
  7. Literal page sentence: complete affirmative solution.
  8. Minimal correction.
  9. Status of the corrected statement.
  10. 6) ADVERSARIAL VERIFICATION
  11. 7) FINAL
  12. 8) COMPLETION ESTIMATE (MANDATORY)
  13. 9) REFERENCES

Erdős Problem #273

1) ROUND-2 OBJECTIVE

Path (C): isolate the exact defect in the statement used in Round-1, prove the literal page sentence completely, and identify the minimal correction needed to recover the genuinely nontrivial problem.

The reason this is the right direction is that Round-1 already showed that the distinct-modulus version is nontrivial and presently unresolved, whereas the literal sentence on the problem page omits the distinctness convention.

2) Round-1 FOUNDATION USED

I use the following Round-1 results.

  1. Round-1 observed that if repeated moduli are allowed, then the modulus $4=5-1$ already gives a trivial cover.
  2. Round-1 proved for the corrected distinct-modulus version that any such cover must have at least $18$ congruences and at least one modulus $\ge 70$.

Only $2$ is carried forward as information about the corrected problem.

3) NEW INSIGHT / TOOL (ROUND-2)

The new point is that the "distinct moduli" correction is not ad hoc. In one of Erdős's own expository sources, a covering system is explicitly written in the form \[ a_i \ (\bmod n_i), \qquad n_1<n_2<\cdots<n_k. \] So the historical convention in that source already bakes in distinctness of the moduli.

Therefore there are really two different statements:

  1. the literal standalone sentence on the webpage, which does not say "distinct", and
  2. the historically intended covering-system question under Erdős's convention, where the moduli are distinct.

4) ATTACK PLAN (ROUND-2)

The exact Round-1 gap was this: it corrected the problem to the distinct-modulus version, but did not yet settle whether that correction was logically forced or merely stylistic.

To close that gap I will:

  1. prove the literal sentence directly by an explicit cover;
  2. justify the minimal corrected statement using Erdős's own distinct-modulus convention;
  3. record that the corrected statement is exactly the open version to which the Round-1 lower bounds apply.

5) WORK (ROUND-2)

Literal page sentence: complete affirmative solution.
Consider the four congruences \[ x\equiv 0 \ (\bmod 4),\qquad x\equiv 1 \ (\bmod 4),\qquad x\equiv 2 \ (\bmod 4),\qquad x\equiv 3 \ (\bmod 4). \] Every integer has one of the four residues modulo $4$, so these four congruences cover all integers.

Their common modulus is $4$, and \[ 4=5-1 \] with $5$ prime and $5\ge 5$. Hence every modulus is of the required form $p-1$ for a prime $p\ge 5$.

So the literal sentence

"Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\ge 5$?"

has the answer Yes, by the explicit cover above.

Minimal correction.
Under Erdős's own distinct-modulus convention, the intended nontrivial statement is:
Does there exist a covering system with pairwise distinct moduli, each of the form $p-1$ for a prime $p\ge 5$?

This is exactly the corrected statement adopted in Round-1.

Status of the corrected statement.
Round-1 already proved two necessary conditions for that corrected problem:
  1. at least $18$ congruences are needed;
  2. at least one modulus must be $\ge 70$.

No construction or impossibility proof is presently known to me beyond those Round-1 obstructions.

6) ADVERSARIAL VERIFICATION

There are two possible objections.

  1. "A covering system already means distinct moduli." If one adopts that convention from the start, then the four congruences modulo $4$ are indeed irrelevant to the intended problem. But that only confirms the need to state the convention explicitly. The standalone webpage sentence does not do so.
  2. "Then the problem has not really been solved." Correct: the corrected distinct-modulus problem remains open. What has been settled in this round is the logical status of the sentence as written: taken literally, it is trivial; taken in Erdős's convention, it becomes the open distinct-modulus problem studied in Round-1.

So no contradiction remains: the defect is a missing convention, and the minimal repair is exactly the distinctness condition.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED).

The literal statement is completely solved in the affirmative by the modulus-$4$ example. The historically intended distinct-modulus version is the minimal corrected statement, and that corrected version remains open. Round-1's lower bounds continue to hold for it.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 65%

9) REFERENCES

  1. P. Erdős, Some of my favourite problems in number theory. The covering-systems section explicitly writes the moduli in strictly increasing order.
  2. T. F. Bloom, Erdős Problem #273. Current webpage statement and current status check.
  3. Round-1 output for Problem #273 (reciprocal-sum obstruction and the lower bounds "at least $18$ moduli" and "some modulus $\ge 70$").