{"id":"3e5ee9ee-403f-4299-badb-94d581612a1f","arxiv_id":"2508.01686","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every integer greater than 23 is conjectured to be expressible as a sum of at most five prime powers p^k with p prime and k >= 2, based on exhaustive and sampled computational evidence.","lead":"This preprint conjectures that every integer greater than 23 can be written as a sum of at most five prime powers, like 2^2 + 3^3 + 5^2. It backs the conjecture with computer checks up to 10^7 and sampled checks near 10^10, making it a testable claim about how flexibly sparse number sets can add up.","discovery_kind":"unclear","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 10^10 sampling evidence is not described in the abstract; without a reproducible protocol, finite checks cannot support a universal statement about all n>23.","rationale":"The reader's weakest assumption is precisely that the computational evidence is reliable and representative, and my stress test agrees: the weak point is the undocumented sample near 10^10, not the absence of a proof, since the paper labels the statement a conjecture. My proposed test would settle whether the large-n support is valid by requiring the full algorithm, the sampling protocol, and an independent reimplementation. Because the full text is not available, the concern cannot be resolved now, so the correct verdict remains UNVERDICTED, which is what the reader already concluded. Thus no verdict adjustment is needed.","tokens_in":808,"tokens_out":6373,"duration_ms":82556,"concrete_test":"Obtain the full text and independently implement the authors' exhaustive algorithm from its pseudocode or code; reproduce the complete verification up to 10^7. Then require the exact sampling protocol for the 10^10 range, including the pseudo-random generator, seed, sample size, and the actual sampled integers. Draw an independent uniform random sample of at least 10^6 integers in [10^7, 10^10] and verify each with a reference branch-and-bound search over prime powers. If every sampled integer requires at most five terms and the authors' protocol is reproducible and unbiased, the concern is resolved; if the protocol is absent, unreproducible, or an independent sample finds an integer needing six or more terms, the conjecture as stated is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a universal assertion over all n>23, but the only evidence beyond 10^7 is described as 'sampling' near 10^10. The abstract gives no algorithm, source code, sample size, seed, or list of sampled integers, so the reader cannot tell whether the samples are representative of all integers in that range or were generated from sums of prime powers, which would make the check circular. Even an honest uniform random sample of bounded size cannot prove the absence of a counterexample beyond the sampled range; it can only make a counterexample less likely. The exhaustive check to 10^7 is stronger, but its correctness depends entirely on the implementation, which is not provided. Since the statement is explicitly a conjecture, the absence of a proof is not itself a defect; the defect is that the primary large-n empirical support is not checkable and may be systematically biased. If the sampling protocol is biased or unreported, the conjecture's main empirical support collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript, as represented by the abstract, proposes a conjecture: every integer n > 23 can be expressed as a sum of at most five prime powers p^k, where p is prime and k is an integer greater than or equal to 2. The support offered is computational: all integers up to 10^7 are claimed to have been checked exhaustively, and specific large numbers up to 10^10 were checked via sampling, with no counterexample found. The paper does not claim a proof, and the abstract frames the statement as a conjecture. The full text is not available for review, so the assessment is based solely on the abstract.","tokens_in":874,"tokens_out":2770,"duration_ms":34619,"significance":"If the conjecture is true, it would be a striking additive basis statement: the sparse set of perfect prime powers would form an asymptotic basis of order five. This would contrast with the sparsity of the set and with the existence of large gaps between prime powers, and it would likely stimulate further work on Waring-type problems for prime powers. The exhaustive verification to 10^7 is, in principle, a valuable empirical data point, provided it is reproducible. The paper is honest in labeling the statement as a conjecture, and the claim is falsifiable. However, as presented, the evidence is not checkable from the abstract, and no heuristic or quantitative argument bridges the gap from finite computation to a universal statement. The novelty of the conjecture is plausible, but the current manuscript does not supply enough detail to assess it.","major_comments":[{"comment":"The conjecture is a universal statement over all integers n > 23, yet the evidence beyond 10^7 consists of an unspecified sampling near 10^10. The abstract gives no algorithm, source code, sample size, sampling rule, or random seed. Without a reproducible protocol, the reader cannot determine whether the sampled integers are representative of all integers in that range or whether they were generated by constructing sums of prime powers, which would make the check circular. This is load-bearing because the claim's support beyond 10^7 rests entirely on this sampling. Please provide a detailed computational protocol, including code or pseudocode, the exact set of sampled integers, and a statement of how they were chosen.","section":"Abstract (central claim)"},{"comment":"A finite computation cannot establish the absence of a counterexample beyond the tested range. The exhaustive check to 10^7 is a genuine lower bound, but no argument is given for why a counterexample should not appear between 10^7 and 10^10 or beyond. If the paper is intended as a conjecture, the evidence should be accompanied by a heuristic estimate or density argument quantifying the likelihood of counterexamples, or at least a clear statement that the claim is an extrapolation. The current formulation gives no quantitative handle on the risk.","section":"Abstract (evidential gap)"},{"comment":"The phrase 'specific large numbers up to 10^10 (via sampling)' is ambiguous and insufficiently precise. It is unclear whether 'up to 10^10' means all integers in an interval near 10^10, a random sample of a certain size, or a hand-picked set of values. The word 'specific' suggests an ad hoc selection, which would not provide representative coverage. Please state exactly which integers were tested, how many, and what selection rule was used.","section":"Abstract (sampling description)"}],"minor_comments":[{"comment":"There is a missing space in '10^7(exhaustively)' in the abstract; it should read '10^7 (exhaustively)'.","section":"Abstract (formatting)"},{"comment":"The phrase 'combinatorial creativity' is informal and undefined. Consider replacing it with a precise description of the additive structure, such as 'the set of perfect prime powers forms an asymptotic basis of order at most five'.","section":"Abstract (terminology)"},{"comment":"The definition of prime powers as p^k with k >= 2 is clear, but the standard term 'prime power' often includes k = 1. The authors should restate this convention in the main text to avoid ambiguity.","section":"Abstract (definition)"},{"comment":"The abstract does not mention related work on Waring's problem, sums of prime powers, or additive bases. A brief contextualization would help readers assess the novelty and relevance of the conjecture.","section":"Abstract (literature)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is currently an abstract-only submission, and the central empirical evidence is not verifiable. The authors should be asked for the full computational protocol, code, and a precise description of the sampling procedure before the paper can be seriously evaluated. The lack of a proof is acceptable for a conjecture paper, but the empirical claim must be checkable. I would also encourage the authors to add a heuristic argument for the plausibility of the conjecture, as this would significantly improve the paper's substance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the abstract of 2508.01686. You get a conjecture: every n > 23 is a sum of at most five prime powers p^k with k ≥ 2. That is a concrete, checkable statement, and it looks new to me, though I can't rule out prior work without the references. The paper does the right thing by labeling it a conjecture and not a theorem. The exhaustive check up to 10^7 is real evidence in that range; I trust a finite check like that more than most things.\n\nThe soft spot is exactly what the stress-test note flags. The sampling near 10^10 is described in two words. No algorithm, no sample size or seed, no list, no code. If the sampled integers were generated as sums of prime powers, the check would be circular and would tell us nothing. I'm not accusing the author of that, but the abstract gives the reader no way to rule it out. Even an honest uniform sample of bounded size only makes a counterexample less likely; it cannot prove absence. So the conjecture rests on the exhaustive 10^7 bound plus luck beyond that. That is fine for a conjecture, but it means the paper's contribution is the conjecture itself, not a proof.\n\nI also find the phrase \"combinatorial creativity\" more sales than math. Harmless, but skip it.\n\nThe main unknowns live in the full text: does it survey Hua's theorem and other Waring-type results? Does it provide code and data? Is the sampling protocol described? Without those, the abstract alone doesn't give enough to judge novelty or soundness. Still, the conjecture is natural and the exhaustive bound gives it a reasonable pedigree. I would want a referee to see the details before rejecting or accepting. My recommendation: send it to review with the clear expectation that the computational artifacts be made available. If the code is missing, that's grounds for rejection.\n\nFor you: if you work on additive bases, the conjecture is a handy target to try to break. I'm not going to cite it in my own work until I see how they sampled.","headline":"A clean conjecture about an additive basis of order five for prime powers, backed by computation we cannot audit from the abstract alone.","tokens_in":1468,"tokens_out":2527,"would_cite":false,"duration_ms":31040,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P32","11P05","11B13"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper conjectures that every integer n > 23 is a sum of at most five prime powers p^k (p prime, k ≥ 2), with exhaustive computational evidence to 10^7 and sampled evidence to 10^10.","keywords":["prime powers","additive basis","additive number theory","computational conjecture","Waring-type problem","prime power sums","integer representation"],"falsifier":"The decisive observation would be a single integer $n > 23$ that cannot be written as a sum of at most five prime powers; an independent exhaustive search to $10^7$ plus targeted sampling near $10^{10}$ and a residue-class obstruction analysis are concrete places to look for it.","tokens_in":510,"feed_emoji":"➕","tokens_out":7210,"duration_ms":75288,"temperature":0.7,"pith_summary":"This paper tries to establish a new additive-basis conjecture: every integer $n > 23$ is a sum of at most five prime powers $p^k$, with $p$ prime and $k \\ge 2$. If true, the set of such prime powers would be an additive basis of order five for all integers beyond 23, even though the set is sparse and contains arbitrarily large gaps. The support offered is computational: exhaustive verification for all integers up to $10^7$ and sampled checks for large values up to $10^{10}$ found no counterexample. The paper does not claim a proof, so the conjecture stands or falls on the quality and representativeness of that finite evidence.","feed_headline":"Five prime powers may cover every integer above 23","feed_subtitle":"A search up to 10^7 finds no number over 23 that needs more than five prime-power summands.","key_machinery":"The central object is the collection of all integers $p^k$ with $p$ prime and $k \\ge 2$, and the conjecture asserts that this collection is an additive basis of order five for every integer greater than 23. The mechanism that carries the argument is computational verification: an exhaustive enumeration of representable numbers up to $10^7$ and a sampling of values near $10^{10}$, both failing to produce a number needing more than five summands. No proof or algorithm is described in the abstract.","core_discovery":"The paper's central claim is the conjecture that every integer $n > 23$ can be written as a sum of at most five terms each of the form $p^k$, where $p$ is prime and $k \\ge 2$. The author states this as a conjecture, not a theorem, and presents computational support: all integers up to $10^7$ were checked exhaustively, and large numbers up to $10^{10}$ were checked by sampling, with no exception found. The significance claimed is that prime powers, despite being sparse and having large gaps, can represent all integers efficiently.","pith_inferences":["The author leaves implicit that the same computational pattern could be tested for exponents $k \\ge 3$, where the set is even sparser, to see whether the five-term bound persists.","A natural next step is to look for modular obstructions: a residue class in which no sum of five prime powers can land would disprove the conjecture in one stroke.","If the conjecture holds, quantitative work such as estimating how many representations each $n$ has becomes a plausible target, even though the present paper only establishes evidence, not estimates."],"forward_implications":["Every integer $n > 23$ would have an explicit representation as a sum of five or fewer prime powers.","The prime powers would form an additive basis of order five exactly on the interval $[24, \\infty)$, a sharper statement than the usual asymptotic basis.","Any future counterexample would have to exceed $10^7$, giving the conjecture a concrete, checkable scope.","The representation function counting sums of five prime powers equal to $n$ would be positive for all $n > 23$."],"supporting_citations":[],"fun_headline_variants":["Conjecture: Every integer over 23 needs at most five prime-power summands","Prime powers conjecture: Five summands cover all integers past 23","Every number above 23 may be five prime powers or fewer","Conjecture: at most five prime powers sum to any integer >23","Five prime powers may suffice for every integer >23"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conjecture rests on the assumption that the finite computational checks through $10^7$ and the sampled values near $10^{10}$ are error-free and representative, so no untested integer requires six or more prime-power summands.","fun_headline_variants_meta":{"raw":{"variants":["Conjecture: Every integer over 23 needs at most five prime-power summands","Prime powers conjecture: Five summands cover all integers past 23","Every number above 23 may be five prime powers or fewer","Conjecture: at most five prime powers sum to any integer >23","Five prime powers may suffice for every integer >23"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000749,"raw_usage":{"total_tokens":3255,"prompt_tokens":787,"completion_tokens":2468,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":403,"completion_tokens_details":{"reasoning_tokens":2373}},"tokens_in":403,"tokens_out":2468,"duration_ms":19533,"temperature":1.0,"reasoning_tokens":2373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:25:37.436710+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The decisive observation would be a single integer $n > 23$ that cannot be written as a sum of at most five prime powers; an independent exhaustive search to $10^7$ plus targeted sampling near $10^{10}$ and a residue-class obstruction analysis are concrete places to look for it.","supporting_citations":[],"review_version":1}