{"id":"0761cd21-f106-4c01-8d74-f2382500fc50","arxiv_id":"2506.10585","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines the Primender sequence, a digit-suffix rule, and reports an informal benchmark showing most tested LLMs fail to infer the rule or generate long valid sequences.","lead":"This paper introduces a simple integer sequence, the Primender numbers, which are primes or numbers with a prime suffix, and uses it to test whether several chatbots can deduce the rule and extend the sequence. It also states a hypothesis about consecutive terms that is a direct consequence of the definitions, and reports that most tested models fail the task.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The benchmark claim collapses under the paper's own methodology: LLM outputs were collected under inconsistent multi-turn prompts with human intervention, then scored with an order-insensitive multiset metric that does not measure sequence generation.","rationale":"The paper is best read as two contributions: a mathematical sequence with a few observations, and an LLM benchmark claiming to rank models on symbolic reasoning. The mathematical parts are mostly correct but thin: the max-delta-5 property is true and easily verified by the last-digit residues 2, 3, 5, and 7, and the computational checks are cheap to reproduce. The proposed hypothesis is essentially tautological: if PEn - LPn = 1, then PEn - 1 = LPn is prime, so LPn is itself a Primender and must be the immediately preceding term, forcing delta = 1. That weakens the 'hypothesis testing' component of the benchmark.\n\nThe load-bearing concern is the evaluation methodology. The central claim—that only ChatGPT o3 inferred the rule and achieved 5.16% error—depends on comparing nine LLMs under inconsistent conditions. The paper itself documents that different models received different numbers of prompts, different amounts of clarification, and in some cases generated their 'output' through code rather than direct token generation. The multiset scoring metric then ignores sequence order, which is the very property a sequence-generation task is supposed to test. The combination of uncontrolled interaction and order-insensitive scoring makes the reported rankings uninterpretable.\n\nA single controlled replication with identical single-turn prompts, no human assistance, exact 100,000-term output, and order-aware scoring would settle whether ChatGPT o3's result is robust. Until then, the benchmark conclusions are unsupported, so the reader's REJECT verdict remains appropriate.","tokens_in":12440,"tokens_out":4851,"duration_ms":60333,"concrete_test":"Re-run the evaluation under a fixed, single-turn protocol: give every model the exact same prompt, allow no clarifying questions or human interventions, fix temperature and decoding parameters, parse outputs directly from raw model text, and require exactly 100,000 consecutive Primender terms beginning at 187. Score the outputs three ways: (1) longest correct ordered prefix, (2) per-position validity of the ordered sequence, and (3) the paper's multiset error as a secondary diagnostic only. If ChatGPT o3 no longer uniquely infers the rule and achieves a low error rate under this protocol, the paper's central benchmark conclusion fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is the comparative LLM result: the claim that only ChatGPT o3 inferred the rule and achieved a 5.16% error rate. That claim requires a controlled evaluation in which all models receive the same task under the same conditions. Section VI and Fig. 8 show this was not the case. ChatGPT o3 was allowed to ask clarifying questions and received answers; ChatGPT 4 used a special 'Think For Longer' feature; Gemini 2.5 Pro's 'generated list' contained only about 2000 numbers when copied, yet it is assigned a 99.46% error rate; DeepSeek's list was produced by the Python program it returned rather than directly by the model; LLaMA required multiple prompt attempts; and several models needed additional prompts to produce downloadable files. These are not equivalent tasks, so the table measures prompt engineering and human intervention as much as symbolic reasoning.\n\nThe scoring metric compounds the problem. The paper explicitly uses an 'Ignore Order Compare Content' algorithm based on collections.Counter, comparing multisets. But the task is to 'generate the next 100,000 terms,' which is an ordered sequence. A multiset score gives full credit for correct numbers regardless of order, so a model that outputs correct terms in a permuted order is not penalized for the central failure. Consequently, the reported 5.16% error rate conflates term recall with sequence inference and cannot support the claim that ChatGPT o3 successfully inferred the correct rule.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Primender sequence, defined as the integers that are prime or have at least one prime suffix (of any length), and proposes it as a benchmark for evaluating symbolic reasoning in large language models. The authors state that the maximum difference between consecutive Primender numbers is 5, and they formulate a hypothesis that whenever PEn - LPn = 1 (where PEn is a Primender number and LPn is the largest prime at most PEn), the gap from the previous Primender number is also 1. They then prompt nine LLMs to infer the rule, evaluate the hypothesis, and generate the next 100,000 terms, reporting that only ChatGPT o3 inferred the rule and achieved a 5.16% error rate under a multiset-based scoring metric. The paper also provides computational verification up to 1,000,000 terms and makes code and data available in a GitHub repository and an OEIS entry.","tokens_in":12753,"tokens_out":9803,"duration_ms":113386,"significance":"If the mathematical claims were rigorously proved and the LLM evaluation were a controlled, apples-to-apples comparison, the Primender sequence could serve as a simple, interpretable symbolic benchmark. The paper does provide reproducible computational artifacts (Python code, generated data, and an OEIS registration), and the central ``max delta 5'' statement is in fact true and admits an elementary proof. However, the manuscript as written has load-bearing flaws: the proof of the main mathematical claim is invalid and outsourced to a non-archival blog, the central hypothesis is a definitional tautology rather than an empirical discovery, and the LLM comparison is neither a controlled experiment nor scored with a metric appropriate for sequence generation. These issues undermine the paper's core claims as they stand.","major_comments":[{"comment":"The hypothesis PEn - LPn = 1 implies Δ = 1 is not an empirical pattern but a direct consequence of the sequence definition. If PEn - LPn = 1, then LPn = PEn - 1, and since LPn is prime, LPn is itself a Primender number. Hence the previous Primender before PEn is exactly LPn, so Δ = 1. The authors present this as a discovered conjecture and ask LLMs to evaluate it, but the problem is circular and does not test meaningful symbolic inference. This should be stated and proved as a one-line lemma, not treated as the centerpiece of a benchmark.","section":"Section IV and Section VI"},{"comment":"The proof that the maximum delta is 5 is invalid as written. Invoking Bertrand's postulate does not show that any six consecutive integers contain a prime, and the disjunction of conditions is asserted without justification. The proof is then referred to the author's own blog [14], which is not an archival source. The claim itself is true and has a simple proof: among any six consecutive integers, at least one has last digit 2, 3, 5, or 7 and therefore qualifies by the unit-digit rule; the bound is attained, for example, by the gap from 47 to 52. The authors should replace the flawed argument with this elementary proof.","section":"Section III.C"},{"comment":"The LLM evaluation is not controlled: models received different numbers of prompting turns, some received answers to clarifying questions (ChatGPT o3), one was run with a special 'Think For Longer' feature (ChatGPT 4), some outputs were produced by Python programs returned by the model rather than directly by the model (DeepSeek R1), one model's copied list contained only about 2000 numbers (Gemini 2.5 Pro), and several required repeated prompts for downloadable output. These are not equivalent tasks, so the reported error rates and the conclusion that only ChatGPT o3 inferred the rule cannot be supported. A benchmark claim requires a fixed protocol applied identically to all models, with all interventions and follow-up prompts documented.","section":"Section VI and Fig. 8"},{"comment":"The scoring metric is inappropriate for the stated task. The task asks models to generate the next 100,000 terms in order, but the Python comparison uses collections.Counter and treats the outputs as multisets, giving full credit for correct numbers regardless of position. A model that outputs all correct terms in a permuted order would receive a 0% error rate under this metric, while a model that outputs only a short prefix is penalized in a way that conflates length, ordering, and rule inference. The reported 5.16% error rate for ChatGPT o3 therefore does not demonstrate correct sequence inference, and the error-rate column in Fig. 8 is not a measure of sequence-generation quality.","section":"Section VI, 'Ignore Order Compare Content'"}],"minor_comments":[{"comment":"The prompt shown to the models already reveals that the sequence is called 'Primender' and hints that the rule involves ending digits; this may artificially guide rule inference and should be acknowledged as a possible confound.","section":"Section V"},{"comment":"The 'Rule Inference Accuracy' column is a binary yes/no with no defined threshold or validation procedure; the basis for assigning 'No' to eight models and 'Yes' to ChatGPT o3 is not stated.","section":"Section VI, Fig. 8"},{"comment":"The sentence 'We encourage scholars to cite and explore the Primender series by the author of this research' is promotional and inappropriate in a research paper; it should be removed.","section":"Section III.D"},{"comment":"References [21] and [22] are identical arXiv identifiers (arXiv:2210.03350), and reference [24] appears to have an incomplete or erroneous author list; these citation problems should be corrected.","section":"References"},{"comment":"There are numerous typos and ungrammatical sentences (e.g., 'hypotheisis', 'gerated', 'it didn't evaluated', 'the pattern found was incorrect'), and large code blocks interrupt the narrative; the code should be moved to an appendix or repository.","section":"General"},{"comment":"The claim that PEn - LPn can take any unit digit except 7 is asserted without proof and only verified empirically; a rigorous argument or a clear statement that it is a conjecture is needed.","section":"Section IV.B"}],"recommendation":"reject","confidential_remarks":"The manuscript's central empirical claim—that ChatGPT o3 uniquely succeeds on this benchmark—rests on an uncontrolled evaluation and a metric that does not measure the stated task. The mathematical 'hypothesis' is a trivial consequence of the definitions, and the proof of the main max-delta statement is invalid and outsourced to a non-archival blog. Although the max-delta claim is true and easily fixable, the benchmark conclusions are not salvageable by minor edits; a complete redesign of the experimental protocol and scoring would be required. I recommend rejection, with the possibility that the authors could resubmit a substantially revised version focusing on the elementary properties of the sequence and a properly controlled LLM evaluation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You asked for my read on the Primender paper. Here it is plainly.\n\nThe new thing is the sequence itself: numbers that are prime or have a prime suffix of any length. That object appears to be new, and the author has put it in OEIS as A384735. Credit is also due for shipping code, a GitHub repo, and a million-term data file; the numerical claim that consecutive Primenders differ by at most 5 is true, and the author verified it computationally. That part is fine.\n\nBut the scientific payload is thin. The main hypothesis — if PE_n − LP_n = 1 then Δ = 1 — is not a discovered pattern; it is a direct consequence of the definitions. LP_n is prime, hence a Primender, and it is less than PE_n; since PE_n = LP_n + 1, there is no integer between them, so the previous Primender is LP_n and the gap is 1. That is a tautology, not a testable conjecture. The paper presents it as an empirical discovery, which is misleading.\n\nThe proof of max-delta-5 is also sloppy. It cites Bertrand's postulate for a claim that doesn't need it, and the real proof is deferred to a blog. The claim is true and easy to prove by a digit-ending argument, so this is a presentation flaw rather than a fatal one.\n\nThe bigger problem is the LLM evaluation. The stress-test note is accurate: models received different numbers of prompts, some had human clarification, one output was produced by a Python program it returned, another's \"copyable block\" contained only about 2000 numbers, and the scoring metric ignores order entirely. The task is to generate the next 100,000 terms in order; using a multiset comparison means a model that emits the right numbers in the wrong order is penalized only for missing/extra items, not for failing at sequence generation. The headline claim that only ChatGPT o3 inferred the rule and achieved 5.16% error is therefore not supported by the evidence in the paper. It measures prompt engineering and human intervention at least as much as symbolic reasoning.\n\nWho gets value from this? Someone looking for a new toy sequence for recreational mathematics or a simple benchmark for informal LLM tinkering might find it amusing. As a research contribution, it does not clear the bar: the math is trivial, the proof is underdeveloped, and the evaluation is not reproducible in any controlled sense. I would not cite it, and I would not bring it to reading group.\n\nRecommendation: desk reject. If an editor were feeling generous, it could go to a low-tier venue as a \"new sequence with informal experiments\" note, but a serious referee's time is better spent on work with a non-tautological hypothesis or a controlled benchmark.","headline":"A good-faith write-up of a toy sequence whose central hypothesis is a tautology and whose LLM benchmark is too uncontrolled to support any comparative conclusion.","tokens_in":795,"tokens_out":1577,"would_cite":false,"duration_ms":57203,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A41","11B83"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Primender sequence, defined as numbers that are prime or end in a prime suffix, is proposed as an interpretable LLM symbolic-reasoning benchmark, with the paper reporting that only ChatGPT o3 inferred its rule at 5.16% error.","keywords":["Primender sequence","prime numbers","integer sequences","symbolic reasoning","large language models","rule inference","hypothesis testing","sequence generation"],"falsifier":"Search the sequence past one million terms: finding two consecutive Primender numbers with gap at least 6, or a term with $\\mathrm{PE}_n - \\mathrm{LP}_n = 1$ whose predecessor is not $\\mathrm{PE}_n - 1$, would refute the structural claims. For the benchmark claim, run all models under identical prompts with no human steering; if ChatGPT o3's error is not near 5.16% or another model infers the rule, the reported ranking is not reproducible.","tokens_in":12218,"feed_emoji":"🔢","tokens_out":11323,"duration_ms":132089,"temperature":0.7,"pith_summary":"The paper introduces the Primender sequence: integers that are prime or whose decimal representation ends in a prime block (last 1, 2, 3, ... digits). It claims the sequence has a deterministic structure—consecutive terms never differ by more than 5, and whenever a term is one more than the largest prime below it, the previous term is exactly one less. The author's purpose is to turn this construct into a symbolic benchmark: give an AI model the first 100 terms, ask it to infer the rule, judge the hypothesis, and generate the next 100,000 terms. Across nine chatbots, only ChatGPT o3 is reported to have inferred the rule, validating the hypothesis and producing the sequence with a 5.16% error; most models failed rule inference or could not produce a complete list. If the benchmark is sound, it offers a reproducible way to separate rule discovery from hypothesis checking in current AI systems.","feed_headline":"One LLM inferred the rule behind this new prime-suffix sequence","feed_subtitle":"A nine-chatbot test of a new integer sequence found only ChatGPT o3 decoded it, with 5.16% error.","key_machinery":"The load-bearing object is the inclusion rule: $n$ is a Primender number exactly when $n$ is prime or one of its decimal suffixes of length 1, 2, 3, ... is prime. This rule is what makes the sequence deterministic yet hard to state from examples, and it is also what powers the structural claims—the $\\Delta\\le5$ bound and the $\\mathrm{PE}_n-\\mathrm{LP}_n=1\\Rightarrow \\Delta=1$ hypothesis. The evaluation machinery is a structured prompt showing the first 100 terms and asking each model to identify the rule, evaluate the hypothesis, and generate the next 100,000 terms; outputs are scored with a multiset error comparing the generated list against the true sequence.","core_discovery":"The central claim is that a simple hybrid rule—$n$ belongs to the Primender sequence iff $n$ is prime or $n \\bmod 10^k$ is prime for some $k\\ge1$—generates a sequence with stable, checkable structure and that this structure makes a good hidden-rule test for LLMs. The paper reports two structural facts: the gap $\\Delta$ between consecutive Primender terms is never larger than 5, and whenever $\\mathrm{PE}_n - \\mathrm{LP}_n = 1$ (the $n$-th term is one more than the largest prime ≤ it), the gap from the previous term is 1. These facts are verified computationally on the first 100,000 and 1,000,000 terms. On the benchmark, the paper reports that only ChatGPT o3 inferred the rule from the first 100 terms, judged the hypothesis correctly, and generated the next 100,000 terms with a multiset error of 5.16%, while the other eight models failed at rule inference, produced high-error lists, or returned no usable list.","pith_inferences":["The paper leaves implicit that the hypothesis is a definitional consequence: if $\\mathrm{PE}_n = \\mathrm{LP}_n + 1$, then $\\mathrm{LP}_n$ is itself prime and hence a Primender term, and because it is the integer immediately below $\\mathrm{PE}_n$, the previous term in the sequence must be $\\mathrm{LP}_n$, so $\\Delta = 1$ necessarily.","The gap bound $\\Delta \\le 5$ is also forced just by the single-digit endings 2, 3, 5, 7: among any five consecutive integers one ends in one of those digits, so a gap of 6 would be impossible even without considering longer prime suffixes or primality.","A sharper benchmark would exclude single-digit-qualified numbers (require the qualifying suffix to have length at least 2, or require the whole number to be prime), forcing models to attend to the modular-suffix structure rather than the common last-digit residue.","A testable extension: give every model a mandatory self-ask turn before generating terms; if the paper's explanatory-questioning explanation is right, models that previously failed rule inference should show substantially lower error."],"forward_implications":["The Primender sequence becomes a ready-made benchmark: any evaluator can hand a model the first 100 terms and check rule inference, hypothesis validation, and 100,000-term extrapolation against one deterministic list.","Because the reported gap is never above 5, generated sequences can be checked cheaply: any output containing a gap of 6 or more is immediately known to be wrong, independent of the full rule.","The result that most models validated the hypothesis but could not infer the rule suggests the benchmark isolates inductive discovery from deductive checking, a distinction central to symbolic-reasoning evaluation.","The observation that the one successful model asked clarifying questions points to prompt designs that encourage self-questioning before sequence generation; the paper links this to Self-Ask and related techniques.","The sequence is infinite and densely packed, so the benchmark can scale to arbitrarily long extrapolation tasks without the rule becoming obvious from the length of the sample alone."],"supporting_citations":[{"why":"Supplies the Python code, the million-number dataset, and the raw chat transcripts used to verify the structural claims and score the LLM outputs; the central computations trace to this repository.","marker":"[10]"},{"why":"Bertrand's postulate is the cited basis for claiming that short integer blocks always contain a prime, the step the paper uses to argue the maximum gap cannot exceed 5.","marker":"[13]"},{"why":"The blog entry is where the paper says the full induction proof of the maximum-delta claim is presented; the paper points to it for the proof rather than giving the induction itself.","marker":"[14]"},{"why":"The online integer-sequence catalog entry A384735 catalogs the Primender sequence as a new integer sequence, anchoring the claim of novelty and giving the benchmark an external reference.","marker":"[15]"},{"why":"The comparative LLM analysis is used as the external baseline the paper aligns with when interpreting why GPT-4-based models outperformed others in symbolic reasoning.","marker":"[19]"},{"why":"The Self-Ask prompting result is invoked to explain the paper's finding that the model which asked clarifying questions performed best.","marker":"[22]"}],"fun_headline_variants":["ChatGPT o3 cracks prime-suffix sequence that stumped 8 other LLMs","New 'Primender' sequence exposes LLM reasoning gap: only o3 got it","Prime-suffix puzzle: 1 of 9 LLMs inferred the rule","Only ChatGPT o3 passed the Primender sequence test","New prime-suffix rule: 8 LLMs flunk, o3 nails it"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusions about model reasoning stand on treating each model's single chat session—with different numbers of clarifying prompts, follow-up requests, and human interventions—as a fair and comparable measure of symbolic inference, and on scoring generated lists as multisets even when a model produced fewer, extra, or differently ordered numbers.","fun_headline_variants_meta":{"raw":{"variants":["ChatGPT o3 cracks prime-suffix sequence that stumped 8 other LLMs","New 'Primender' sequence exposes LLM reasoning gap: only o3 got it","Prime-suffix puzzle: 1 of 9 LLMs inferred the rule","Only ChatGPT o3 passed the Primender sequence test","New prime-suffix rule: 8 LLMs flunk, o3 nails it"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000622,"raw_usage":{"total_tokens":2952,"prompt_tokens":1084,"completion_tokens":1868,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":700,"completion_tokens_details":{"reasoning_tokens":1766}},"tokens_in":700,"tokens_out":1868,"duration_ms":13541,"temperature":1.0,"reasoning_tokens":1766,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:22:53.924664+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search the sequence past one million terms: finding two consecutive Primender numbers with gap at least 6, or a term with $\\mathrm{PE}_n - \\mathrm{LP}_n = 1$ whose predecessor is not $\\mathrm{PE}_n - 1$, would refute the structural claims. For the benchmark claim, run all models under identical prompts with no human steering; if ChatGPT o3's error is not near 5.16% or another model infers the rule, the reported ranking is not reproducible.","supporting_citations":[{"cited_title":"Github repository for Primender,","cited_arxiv_id":null,"evidence_quote":"Supplies the Python code, the million-number dataset, and the raw chat transcripts used to verify the structural claims and score the LLM outputs; the central computations trace to this repository."},{"cited_title":"A proof of Bertrand’s postulate,","cited_arxiv_id":null,"evidence_quote":"Bertrand's postulate is the cited basis for claiming that short integer blocks always contain a prime, the step the paper uses to argue the maximum gap cannot exceed 5."},{"cited_title":"Primender Sequence: Numbers that Are Prime or End in Prime,","cited_arxiv_id":null,"evidence_quote":"The blog entry is where the paper says the full induction proof of the maximum-delta claim is presented; the paper points to it for the proof rather than giving the induction itself."},{"cited_title":"A384735: Primender Sequence: Numbers that are prime or end in a prime number (of any length),","cited_arxiv_id":null,"evidence_quote":"The online integer-sequence catalog entry A384735 catalogs the Primender sequence as a new integer sequence, anchoring the claim of novelty and giving the benchmark an external reference."},{"cited_title":"Comparative Analysis of Top LLMs: GPT-4, Gemini, Claude and More,","cited_arxiv_id":null,"evidence_quote":"The comparative LLM analysis is used as the external baseline the paper aligns with when interpreting why GPT-4-based models outperformed others in symbolic reasoning."}],"review_version":1}