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Math-5456-A2 · revision r1 · on Math-5456

gpt pro 5.2 attempt on Math-5456

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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. Problem 141: consecutive primes in arithmetic progression
  2. 1) FORMAL RESTATEMENT
  3. 2) QUICK LITERATURE / CONTEXT CHECK
  4. 3) ATTACK PLAN
  5. 4) WORK
  6. 4.1 A necessary divisibility condition on the common difference
  7. 4.2 Explicit examples for small $k$
  8. 5) VERIFICATION / EDGE CASES
  9. 6) FINAL

Problem 141: consecutive primes in arithmetic progression

1) FORMAL RESTATEMENT

Let $p_1=2<p_2=3<p_3=5<\cdots$ be the increasing sequence of primes. Fix an integer $k\ge 3$.

Definition (Consecutive-prime arithmetic progression, CPAP-$k$): A CPAP-$k$ is an index $n$ and a step $d\ge 1$ such that \[ (p_n,p_{n+1},\dots,p_{n+k-1}) = (a, a+d, \dots, a+(k-1)d) \] for some integer $a$. Equivalently, $p_{n+i}=a+id$ for all $i=0,\dots,k-1$.

The question is: for each fixed $k\ge 3$, does there exist a CPAP-$k$? A stronger open question (mentioned in the prompt) is whether there are infinitely many CPAP-$k$ for fixed $k$ (open even for $k=3$).

2) QUICK LITERATURE / CONTEXT CHECK

  • Green--Tao proved that the primes contain arbitrarily long arithmetic progressions, but their progressions need not be consecutive primes.(See [GreenTao2008] and background summaries such as [ErdosProblems141].)

  • The "consecutive" requirement is much stronger and is open in general. Computational searches have exhibited examples for small $k$ (the prompt mentions verification for $k\le 10$; this is often recorded under the name CPAP-$k$).(See [WikiCPAP,T5KCPAP].)

  • Even for $k=3$, infinitude is open: $p_n,p_{n+1},p_{n+2}$ are in arithmetic progression iff consecutive prime gaps satisfy $p_{n+1}-p_n = p_{n+2}-p_{n+1}$.

3) ATTACK PLAN

There are two natural directions.

  1. Obstruction search. Prove necessary congruence/divisibility constraints on the common difference $d$ of any prime AP of length $k$. This can show such progressions must be extremely sparse, but does not by itself rule them out.

  2. Existence/infinitude via prime patterns. Heuristically, Hardy--Littlewood type conjectures for prime $k$--tuples suggest infinitely many CPAP-$k$, but proving even CPAP-$3$ infinitude would require breakthroughs on prime gaps/patterns far beyond current technology.

In the "work" below I (i) prove a standard "primorial divides $d$" lemma, and (ii) give explicit small-$k$ examples (with a computational verification of primality and consecutiveness).

4) WORK

4.1 A necessary divisibility condition on the common difference

Lemma (Primorial divisibility unless a small prime appears): Let $k\ge 3$ and let $a,a+d,\dots,a+(k-1)d$ be primes (not necessarily consecutive). Let $q$ be a prime with $q\le k$. If $q\nmid d$, then one of the terms $a+id$ is divisible by $q$; since it is prime, that term must equal $q$. Equivalently: if none of the primes in the progression equals $q$, then $q\mid d$.

In particular, if all $k$ primes in the progression are $>k$, then $d$ is divisible by the primorial \[ \prod_{q\le k\ \text{$q$ prime}} q. \]

Proof: Work modulo $q$. If $q\nmid d$, then $d$ is invertible mod $q$. Hence the residues \[ a,\ a+d,\ a+2d,\dots,\ a+(q-1)d \pmod q \] are all distinct, i.e. they are a complete residue system modulo $q$. In particular, among the first $k\ge q$ terms $a+id$ ($0\le i\le k-1$) there exists some $i$ with $a+id\equiv 0\pmod q$. Thus $q\mid (a+id)$. Since $a+id$ is prime, this forces $a+id=q$. The contrapositive gives: if $q$ does not appear among the terms, then we must have $q\mid d$. ∎
Remark: Lemma Lemma (Primorial divisibility unless a small prime appears) applies in particular to CPAP-$k$. For example, if a CPAP-$11$ exists with all terms $>11$ (the only realistic case), then its common difference is a multiple of $2\cdot 3\cdot 5\cdot 7\cdot 11=2310$. This matches the folklore statement that CPAP-$11$ would require a very large step size.(See [T5KCPAP] for this specific observation in the CPAP context.)

4.2 Explicit examples for small $k$

The following explicit CPAP-$k$ examples are recorded in standard tables.(See [WikiCPAP,T5KCPAP].) I also verified (by trial division primality testing and checking for intervening primes) that each listed tuple consists of consecutive primes.

\begin{align*} \text{CPAP-3:}\quad & 3,5,7 && (d=2).\\ \text{CPAP-4:}\quad & 251,257,263,269 && (d=6).\\ \text{CPAP-5:}\quad & 9843019,9843049,9843079,9843109,9843139 && (d=30).\\ \text{CPAP-6:}\quad & 121174811,121174841,121174871,121174901,121174931,121174961 && (d=30). \end{align*}

5) VERIFICATION / EDGE CASES

  • Lemma Lemma (Primorial divisibility unless a small prime appears) has a small-prime exception: if $q\le k$ is itself one of the terms, then $q\nmid d$ is possible (e.g. progressions starting at $2$ can have odd $d$). For CPAP-$k$ with large $k$ this exception is irrelevant because the terms are enormous.

  • The examples in §4.2 are genuinely "consecutive" primes: between successive terms $a+id$ and $a+(i+1)d$ there is no other prime.

6) FINAL

UNRESOLVED

  1. Farthest point reached. I proved the primorial divisibility obstruction (Lemma Lemma (Primorial divisibility unless a small prime appears)) and verified concrete CPAP-$k$ examples for $k\le 6$.

  2. Best partial lemma. Lemma Lemma (Primorial divisibility unless a small prime appears): for any prime AP of length $k$, every prime $q\le k$ divides the common difference unless $q$ appears as a term.

  3. Smallest missing step. To prove existence (or infinitude) of CPAP-$k$ would require controlling not just primes in patterns but also exclusion of primes in the intervals between successive terms (consecutive prime gaps), a problem currently out of reach even for $k=3$.

  4. Completion estimate. A proof seems to require breakthroughs comparable to (or stronger than) major open problems on prime $k$--tuples and structured prime gaps; no known method approaches CPAP-$k$ infinitude.