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ERDőS #472 · PARTIAL

Erdős problem 472 — wave 9m

Outcome

The problem is not solved here. I obtained a reproducible finite construction substantially beyond the public finite checks that I found: starting from \(3,5\), the greedy sequence exists through

\[ N=10{,}000{,}000,\qquad q_N=437{,}662{,}243. \]

This is a (d) computational-only statement, not evidence sufficient to deduce infinitude. In the same certified range every generated gap is at most \(3270\), and this is sharp.

0. Mandatory live-page check

I fetched the live page, its LaTeX view, history, citation popup, and discussion thread through the Bright Data browser on 2026-07-28. Direct cached metadata was not used for the gate.

The current statement, verbatim, is:

Given some initial finite sequence of primes \(q_1<\cdots<q_m\) extend it so that \(q_{n+1}\) is the smallest prime of the form \(q_n+q_i-1\) for \(n\geq m\). Is there an initial starting sequence so that the resulting sequence is infinite?

The page currently says:

“working on formalising” all have value None;

Thus none of the mandatory stop conditions applies.

The page's complete mathematical note is: “A problem due to Ulam. For example if we begin with \(3,5\) then the sequence continues \(3,5,7,11,13,17,\ldots\). It is possible that this sequence is infinite.”

There is one comment, by StijnC at 06:18 on 24 Aug 2025. It says that the \(3,5\) sequence “should work,” gives the explicitly heuristic model

\[ q_{n+1}=q_n+\log(q_n)^{2+o(1)},\quad q_n=n^{1+o(1)},\quad (1-1/\log q_n)^n=\exp(-n^{1-o(1)}), \]

and reports an “easy check” through at least \(10^6\) terms. The page itself warns that comments are unverified. I therefore treat the formulas as (c) plausible/structural-unverified and the million-term assertion as an unverified computational lead until independently recomputed.

The displayed statement omits an explicit range for \(i\). The example and the definition/code in A389713 both give the natural intended reading \(1\leq i\leq n\): candidates use already present terms. That is the interpretation checked below.

1. Exact reformulation

Whenever the next term exists, define its source index

\[ s(n)=\min\{\,i:1\leq i\leq n,\ q_n+q_i-1\text{ is prime}\,\}. \]

Since the \(q_i\) increase, minimizing the candidate is the same as minimizing \(i\), and

\[ q_{n+1}=q_n+q_{s(n)}-1. \tag{1} \]

Put \(a_n=q_n-1\). Subtracting one from (1) gives

\[ a_{n+1}=a_n+a_{s(n)},\qquad a_k+1\text{ prime}. \tag{2} \]

Thus the problem asks for an infinite greedy star-addition chain all of whose shifted terms are prime. This equivalence is (a) elementary-rigorous.

There is also an exact prime-difference identity:

\[ q_{n+1}-q_{s(n)}=q_n-1. \tag{3} \]

Consequently, every even number \(q_n-1\) encountered by a continuing sequence must be a difference of two primes, with the smaller prime required to be a term already in the chain. I will call this the self-feeding Maillet-chain condition. Equation (3), not the terminology, is (a) elementary-rigorous.

Two simple necessary structural facts are useful:

  1. A single source \(j\) cannot be used in \(q_j\) consecutive transitions.

If it were, those new primes would be \(q_n+t(q_j-1)\) for \(1\leq t\leq q_j\). As \(q_j-1\equiv-1\pmod {q_j}\), one is \(0\pmod {q_j}\) and is larger than \(q_j\), a contradiction. This is (a) elementary-rigorous.

  1. In any infinite sequence the source indices are unbounded. If

\(s(n)\leq K\) eventually, then all eventual gaps are at most \(q_K-1\), so \(q_n=O(n)\). This would put a positive linear density of distinct primes in the integers, contradicting \(\pi(x)=o(x)\). This is (b) rigorous-modulo-the-prime-number-theorem. In particular, no argument involving only a fixed finite set of shifts can prove infinitude.

2. Literature search and what it did not find

The live citation popup expands [ErGr80] to:

P. Erdős and R. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980), MR 0592420.

The live page attributes the question to Ulam. I verified the monograph's bibliographic existence through the live citation, MathSciNet identifier, the Erdős publication archive, and the Google Books catalogue. I did not locate an openly readable scan of the complete 128-page monograph, so I do not claim to have checked an original page beyond the authoritative live transcription.

Exact-statement, exact-recurrence, initial-term, MR-number, and citation searches across web/Scholar-indexed results and arXiv found no research paper specific to this recurrence. This is a search miss, not a claim that no such paper exists.

The only sequence-specific external data found was OEIS A389713, submitted 8 Jan 2026. It says that infinitude is unknown and links a table through \(n=10{,}000\). I downloaded that table only for a post-computation comparison. All 10,000 terms match: its little-endian uint32 SHA-256 is 192e51bba0f320bd6e9baecae759ee0e32b8ee5622ee90fc3f3938c2a7ade80c, and its independently calculated FNV checkpoint is 184654103687052557, exactly the recomputed value below.

For the surrounding prime-difference obstruction, two verified primary sources are relevant:

primes*](https://arxiv.org/abs/1206.0149), arXiv:1206.0149 (2012), studies gaps between integers representable as differences of two primes and proves that every sufficiently large interval \([X,X+X^\epsilon]\) contains such an integer.

primes*](https://arxiv.org/abs/1505.00187), arXiv:1505.00187v3 (2015), proves a strong largeness property (\(\Delta_r^*\)) for that difference set.

Neither result says that each particular adaptively selected \(q_n-1\) has a representation, much less one whose smaller prime lies in \(\{q_1,\ldots,q_n\}\). Pintz's introduction records Maillet's still unproved conjecture that every even integer is a difference of two primes.

3. Exact computation

Algorithm and why it is an exact check

For a requested terminal index \(N\), an Eratosthenes sieve is built through the claimed \(q_N\). At a transition, candidates

\[ q_n+q_1-1<q_n+q_2-1<\cdots \]

are tested in that order and the first prime is appended. Every rejected candidate is smaller than the accepted \(q_{n+1}\leq q_N\), so a sieve only through \(q_N\) covers every primality decision made in the entire run. Eratosthenes is deterministic: every composite at most the limit has a prime factor at most its square root.

The core of the standalone checker is:

flags = bytearray([1]) * ((prime_limit >> 1) + 1)
flags[0] = 0
for p in range(3, math.isqrt(prime_limit) + 1, 2):
    if flags[p >> 1]:
        start = (p * p) >> 1
        count = ((len(flags) - 1 - start) // p) + 1
        flags[start::p] = b"\x00" * count

q = array("I", [3, 5])
while len(q) < target:
    current = q[-1]
    for i, source in enumerate(q, 1):
        candidate = current + source - 1
        if flags[candidate >> 1]:
            q.append(candidate)
            break
    else:
        raise AssertionError("the sequence terminated")

The complete independent checker is erdos472_wave9m_reverify.py. It additionally verifies every checkpoint, source index, rejected candidate, the addition-chain identity, the maximum-gap witness, two rolling hashes, and a SHA-256 digest. It uses only the Python standard library.

I first computed the data with the separate packed-bit C++ implementation erdos472_wave9m_scan.cpp. The Python implementation uses a different memory representation and recomputes the complete sequence rather than reading C++ output.

Certified table

Let \(S_N=\max_{2\leq n<N}s(n)\). The “decisions” column is the exact total number of candidate primality tests, including accepted candidates.

| \(N\) | \(q_N\) | \(S_N\) | sharp max gap \(q_{S_N}-1\) | decisions | |---:|---:|---:|---:|---:| | 10 | 31 | 3 | 6 | 13 | | 100 | 631 | 10 | 30 | 278 | | 1,000 | 12,829 | 34 | 156 | 4,695 | | 10,000 | 190,669 | 70 | 420 | 63,257 | | 100,000 | 2,678,747 | 100 | 630 | 813,646 | | 1,000,000 | 35,009,411 | 179 | 1,428 | 9,929,165 | | 10,000,000 | 437,662,243 | 325 | 3,270 | 117,195,907 |

All table entries are (d) computational-only.

The terminal maximum is witnessed for the first time by

\[ \begin{aligned} q_{7{,}956{,}666}&=341{,}181{,}713,\\ q_{7{,}956{,}667}&=341{,}184{,}983 =q_{7{,}956{,}666}+q_{325}-1,\\ q_{325}&=3271. \end{aligned} \]

Hence the gap is exactly \(3270\), proving sharpness within the certified finite range. This is (d) computational-only.

The SHA-256 of the complete little-endian uint32 stream \((q_1,\ldots,q_{10^7})\) is

796b4a09e289287fccf6b0ad9b9766ac89e27d419253a0dae3901e8305924214

The final independent Python audit took 75.38 wall seconds and 372,932 KiB maximum resident memory on this VM. It ended with:

PASS checkpoint n=10,000,000 q_n=437,662,243 max_source=325
     max_gap=3270 candidates=117,195,907
PASS exact greedy recurrence through n=10,000,000; q_n=437,662,243

Reproduction commands:

python runs/erdos472_wave9m_reverify.py --quick
python runs/erdos472_wave9m_reverify.py

g++ -O3 -std=c++17 -Wall -Wextra -pedantic \
  runs/erdos472_wave9m_scan.cpp -o /tmp/erdos472_scan
/tmp/erdos472_scan 10000000 437662243

4. Exact remaining wall

At step \(n\), the precise missing assertion is

\[ \sum_{i=1}^{n}\mathbf 1_{\mathbb P}(q_n+q_i-1)>0. \tag{4} \]

The set of shifts in (4) is sparse, generated by the earlier successes, and correlated with the current prime. A lower-bound sieve would have to show that this adaptive set contains a prime for every \(n\), not merely for almost all \(n\) or on average. This is exactly where the parity obstruction to ordinary sieve lower bounds appears.

Even after discarding the self-feeding restriction and allowing an arbitrary prime as the smaller member, the pointwise statement that every even integer is a prime difference is Maillet's open conjecture. Known “syndetic” or short-interval results ensure that some nearby even integers are prime differences; they do not force the specific integer \(q_n-1\). Restoring the requirement that the smaller prime be one of the previous \(q_i\) is strictly more restrictive.

Thus the exact uniform lemma needed to close the positive direction is (4), or a stronger theorem implying (4) for this self-generated chain. A negative solution would require an actual \(n\) for which all \(n\) candidates in (4) are composite. No finite survival bound, including \(10^7\), supplies the uniformity or a termination witness. This is a WALL, not a conjectural proof of infinitude.

PARTIAL: The seed \(3,5\) is exactly certified through \(10{,}000{,}000\) terms with \(q_{10^7}=437{,}662{,}243\) and sharp gap bound \(3270\) on that range; infinitude remains open because the required adaptive prime lower bound (4) is unavailable.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger