{"id":"ea72886d-a1fa-4413-86c6-91898675c456","arxiv_id":"2506.11429","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"This survey catalogs ideal numerical solutions for 296 PTE/GPTE/FPTE types and proposes, without complete proof, generalized Newton identities and six conjectural bounds to speed up computer searches.","lead":"This paper surveys the Prouhet-Tarry-Escott problem, a classic Diophantine question about two sets of numbers whose sums of powers match, along with generalized versions. It catalogs hundreds of solutions and proposes new algebraic identities and six conjectural search bounds, but leaves the new claims unproved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Identity 4 is unproved and drives Chapter 5's exact-bound derivation; if the determinant claims used in Example 5.2 fail, the β_6,min polynomial and the accelerated search are unsupported.","rationale":"Reader's weakest_assumption is correct and coincides with my read: the unproved Identity 4 is the fragile engine. I verified the concern by checking the text: Section 2.2.2 admits no complete proof, and Example 5.2 explicitly says the polynomial is obtained by applying Identity 4. The survey component is valuable and well-referenced, and the six conjectures are honestly labelled as conjectures. The main risk is that Identity 4 is asserted as an identity, not a conjecture, while its proof is missing. A direct elimination test would settle whether the specific derivation in Example 5.2 is correct; if it matches, the conditional accept is appropriate given the explicit labelling. Therefore no change to the reader's CONDITIONAL verdict.","tokens_in":103387,"tokens_out":14631,"duration_ms":148769,"concrete_test":"Use a computer algebra system to re-derive Example 5.2's equation (5.30) from the system (5.27) by direct variable elimination (resultant or Gröbner basis) without invoking Identity 4. If the minimal polynomial for y_6 differs from 512 y_6^6 - 256 y_6^5 - 1504 y_6^4 + 1968 y_6^3 - 830 y_6^2 + 120 y_6 - 5 = 0, then the Chapter 5 bound derivation is unsound; if it matches, the specific load-bearing application of Identity 4 is supported (though the general proof gap remains).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Identity 4 (Section 2.2.2) is the load-bearing algebraic tool: the paper states 'It is challenging for us to fully prove Identity 4' and offers only Mathematica checks for small n, yet Chapter 5 uses it as a fact. In Example 5.2, the derivation of the exact lower bound β_6,min via the polynomial (5.30) proceeds by asserting that the 3×3 determinants (5.34) and (5.35) vanish 'according to Identity 4'. These vanishing conditions are not labelled conjectural; they are used to solve for the unknown P5 and then for y6. Because the same method is applied in Examples 5.3–5.6, including cases with k=0 and negative k, a hidden restriction or failure of Identity 4 in any of these regimes would make the claimed exact bounds incorrect rather than merely unproved. The GPTE surveys and numerical catalog do not depend on this, but the new search-acceleration method does.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript is a survey of the Prouhet-Tarry-Escott problem, its generalized forms (GPTE and FPTE), multigrade chains, and trigonometric extensions. Its central new mathematical content consists of three generalizations of the Girard-Newton identities (Identities 3, 4, and 5), a set of constant identities (Identities 6--9), trigonometric identities (Identities 10--14), and a normalized GPTE framework with six conjectures giving claimed exact bounds for every normalized variable, proposed as a tool for accelerating computer searches. A large appendix catalogs 296 types of ideal GPTE and FPTE solutions, and a reference search program is described.","tokens_in":103673,"tokens_out":2644,"duration_ms":36237,"significance":"If the three generalized Girard-Newton identities and the six normalized-GPTE conjectures were proved, the manuscript would make a substantial contribution to the PTE/GPTE literature: it would provide new structural tools, a systematic bound framework that could accelerate searches, and a large organized catalog of ideal solutions. The survey portions and the extensive appendix have independent utility as a reference. The paper is also commendable for including explicit numerical examples, Mathematica verification statements, and reference code. However, the load-bearing new results are explicitly unproved, and both the exact-bound claims and the search-acceleration method rest on them, so the mathematical claims as currently presented are not yet established.","major_comments":[{"comment":"Identity 4, stated in Section 2.2.2 as holding for all positive integers n and m and all integers k, is used as an established fact in Chapter 5, although the text says 'It is challenging for us to fully prove Identity 4' and only Mathematica checks for small n are provided. In Example 5.2, the derivation of the polynomial (5.30) for the exact lower bound beta_6,min proceeds by asserting that the determinants (5.34) and (5.35) vanish 'according to Identity 4'. Because the same method is used in Examples 5.3 through 5.6, including cases with k=0 and negative k, a hidden restriction or failure of Identity 4 in any of these regimes would make the claimed exact bounds incorrect rather than merely unproved. This is load-bearing for the central claim of Chapter 5, so Identity 4 must either be proved or re-stated with precise hypotheses and an independent verification for every regime used afterward.","section":"Section 2.2.2 and Example 5.2"},{"comment":"Identity 3 is also explicitly unproved, yet it is used to derive the constant identities in Chapter 3 and to justify several 'no nontrivial solution' corollaries such as Corollaries 2.8 and 2.9. The paper states, 'At present, we are unable to provide a complete mathematical proof for Identity 3.' A derivation that relies on an unproved identity should be marked as conditional, and the corollaries that exclude the existence of solutions should be labeled as conditional statements until a proof is supplied. In particular, the proof ideas for Corollaries 2.10 and 2.11 depend on the validity of (2.98), which is a special case of Identity 3.","section":"Section 2.2.1 and Chapter 3"},{"comment":"The six conjectures in Chapter 5 are used to define the 'exact bounds' of the normalized GPTE system and to derive the search acceleration described in Chapter 6, but the paper does not prove attainability or infimum status of the collapsed configurations in Conjecture 2. The positive infinitesimal epsilon_1 is introduced in (5.22) and later replaced numerically by 10^-50 in Examples 5.4--5.6; the paper does not justify that the limit as epsilon_1 tends to 0 is the actual lower bound for integer solutions, nor whether the bound is attained or only approached. The statements would be cleaner if the paper distinguished rigorously between the infimum of the normalization and the minimum over integer solutions, and if it stated exactly which claims are conditional on the conjectures.","section":"Section 5.1 and Section 5.2"},{"comment":"The search algorithm in Chapter 6 is presented as a method for finding ideal non-negative integer solutions, but its completeness claim depends on the unproved bounds conjectures. If only some of the conjectures hold, the algorithm still finds a subset of solutions, yet the exposition repeatedly refers to 'exact bounds' and 'the theoretical lower bound' (for example, equations (5.29)-(5.30)) without indicating that these values are conditional. Section 5.2 should explicitly separate the conditional derivation of the bounds from any unconditional verification of the resulting solutions, and the description of the search algorithm should state that it is heuristic with respect to completeness unless the conjectures are proved.","section":"Chapter 6 and the search algorithm"}],"minor_comments":[{"comment":"In equation (2.151), the last displayed identity reads 'W12 = (a1a2) · W12', which is self-referential and cannot be correct; it should presumably read 'W12 = (a1a2) · W10'. Please correct this typo.","section":"Section 2.2.2, Example 2.10"},{"comment":"The text contains several small language and typographical errors, including 'For for all positive integers k' in Example 2.3, 'Exapmle' in Example 2.5's header reference, and 'by applying Identity 4' in Example 5.2 where the identity is unproved; a careful editing pass would improve readability.","section":"Throughout"},{"comment":"The statement of the third generalization, Identity 5, is complex and is said to be verified only 'by using Mathematica for a certain smaller n'; a precise statement of the range of n for which it is known to hold, and for which it is only conjectured, would help the reader use the many examples in Section 2.2.3.","section":"Chapter 2, discussion of Identity 5"},{"comment":"The appendix is extremely long and is presented as a 'Collection of ideal numerical solutions' with cross-references, but there is no explicit statement of which entries were independently verified and which are taken from the author's prior work; adding a verification column or a provenance note would be useful.","section":"Appendix and references"}],"recommendation":"major_revision","confidential_remarks":"The manuscript would fit a survey or computational number theory venue if the main new results are presented as conditional on a small set of clearly stated conjectures, or if the missing proofs are supplied. The self-citation pattern is heavy, and the novelty of Identities 3--5 relative to the author's previous papers should be clarified in the revision. The stress-test concern about Identity 4 is substantiated by the text: the identity is used as a fact in Example 5.2 and throughout Chapter 5 despite the paper's explicit statement that it is not fully proved. I would recommend major revision, with the understanding that the survey and appendix portions may be publishable even if the generalized identities remain conjectural, provided their status is made unambiguous."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the survey half is genuinely worth having; the advertised new machinery is not yet at research-claim strength. Identity 4 is the load-bearing algebraic tool, the paper says it cannot fully prove it, and Example 5.2 then uses it as a fact to derive the exact lower-bound polynomial for beta_6. That is the soft spot to focus on.\n\nWhat the paper does well: it organizes a large body of ideal GPTE/FPTE solutions, gives explicit credit to earlier authors (Choudhry, Gloden, Wróblewski, etc.), and makes the mirror-type transformations easy to follow. The 296-type appendix and the 2023/2025 numerical solutions are concrete, checkable data. The three generalized Girard-Newton identities are genuinely new in form, even if they are self-cited and unproved. The normalized GPTE setup and the six conjectures are a plausible program for accelerating searches; they are labeled as conjectures in Chapter 5, which is honest.\n\nThe problems are in proportion: the central new content rests on unproved identities. The stress-test note holds up. Identity 4 is stated for all positive n, m and all integer k, but the proof is only \"challenging\" and Mathematica checks are for small cases. In Example 5.2, the vanishing of the 3x3 determinants (5.34) and (5.35) is asserted \"according to Identity 4,\" and that is how the polynomial for beta_6,min is obtained. The same method is used for examples with k=0 and negative k. If Identity 4 has hidden restrictions, the exact bounds and the search-acceleration method do not follow. This is not a manufactured flaw; it is the load-bearing step. The six conjectures are also just that, and the abstract's claim that the approach \"enhances efficiency\" would be fairer if it said \"proposed bounds, if valid, would enhance efficiency.\" The infinitesimal epsilon_1 in the lower-bound derivations is a numerical convenience; the limit needs a proof.\n\nThe survey value is independent of those gaps, and the catalog can stand on its own. My recommendation: send it to peer review, but with the expectation of heavy revision -- either supply proofs for Identities 3-5 or relabel every unproved identity as a conjecture throughout, soften the abstract, and ship the search code in a verified repository. A serious referee in equal-sums-of-like-powers would find this paper worth engaging with, especially for the survey and numerical data.","headline":"A useful survey and numerical catalog fronted by an unproved identity that Chapter 5 leans on as a theorem.","tokens_in":104099,"tokens_out":2048,"would_cite":true,"duration_ms":29012,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D09","11D25","11D41"],"pacs":[],"model":"deepseek-v4-flash","headline":"Three new generalizations of the Girard-Newton identities make the classic equal-sums-of-like-powers problem systematically solvable.","keywords":["Prouhet-Tarry-Escott problem","Equal sums of like powers","Girard-Newton identities","Diophantine equation","Multigrade chains","Generalized PTE problem","Normalized GPTE problem","Ideal solutions"],"falsifier":"The concrete test is a direct spot-check of Identity 4 outside the small cases the paper says it verified symbolically: choose $n=4$, $m=5$, include several negative exponents and the zero-exponent definition of $P_0$, pick random integer tuples with no zero entries and no coincidences $a_i=b_j$, and check numerically whether $W_{20}=\\prod_{i=1}^{4}\\prod_{j=1}^{5}(a_i-b_j)$; one mismatch falsifies the identity as stated, and a pattern of mismatches at zero entries would reveal the missing restriction. A second test targets the search engine itself: recompute the polynomial for $\\beta_{6,\\min}$ of type $(k=1,2,3,4,6)$ exactly as in Example 5.2 by applying Identity 4, and compare its root to a high-precision numerical solution of the collapsed system (5.27); any discrepancy between the two routes refutes the bound conjecture that the search algorithm relies on.","tokens_in":103114,"feed_emoji":"🔢","tokens_out":20057,"duration_ms":190229,"temperature":0.7,"pith_summary":"This paper claims that the Prouhet-Tarry-Escott problem — two distinct sets of integers with equal sums of like powers — and its two generalizations can be handled by three new algebraic tools derived from the classical Girard-Newton identities. The first tool extends the identities to all integer exponents, including negative powers and the equal-products case $k=0$; the second packages them into determinants that are claimed to equal the product of all pairwise differences between the two sets; the third handles odd exponents. The paper also introduces a normalized version of the generalized problem, scaled so the largest entry is $1$, and states six conjectures giving exact upper and lower bounds for every variable one by one, which the author reports using to accelerate computer searches for ideal solutions. A sympathetic reader would care because, if the identities and the bounds hold, entire families of these classical Diophantine problems become routinely searchable, and the paper offers what it describes as the most complete census to date: 296 types of generalized and Fermat-form problems with ideal solutions. The paper is candid that complete proofs of the identities and conjectures are not yet given; the supporting evidence is symbolic verification in Mathematica for small parameter values and agreement with known solutions.","feed_headline":"Equal-powers problem yields to three new identities","feed_subtitle":"Girard-Newton generalizations plus six bound conjectures let computers find ideal solutions, the survey argues.","key_machinery":"The central mechanism is the determinant family of Identity 4. For $n$ numbers $a_i$ and $m$ numbers $b_j$ one forms the power-sum differences $P_k$, with $P_0$ defined as $-\\prod_j b_j/\\prod_i a_i$, builds an auxiliary sequence $T_k$ by the Newton recursion on both positive and negative indices, and then forms determinants $W_{nk+r}$ whose rows are consecutive shifts of the sequence $S_k$; their defining property is $W_{nm}=\\prod_{i=1}^{n}\\prod_{j=1}^{m}(a_i-b_j)$ and $W_{nk+r}=e_r(a_1,\\dots,a_n)\\,W_{nk}$, where $e_r$ is the $r$-th elementary symmetric sum. Since an ideal solution forces the full difference product to vanish, the identity converts equalities of power sums into the statement that certain ratios $W_{nk+r}/W_{nk}$ are small integers with alternating signs; a search loop tests candidates by these ratio checks and aborts the moment a ratio is not an integer, which the paper reports being much faster than checking the power sums directly. Identity 3 ($T_k=0$ for all integer $k$) does the complementary work of excluding whole exponent sets, and Identity 5 supplies the odd-exponent analogue. The second engine is the normalized-GPTE framework of Chapter 5: after scaling the largest entry to $1$, six conjectures assert that the two ordered sets interlace and that exact bounds for $\\beta_{n+1}$, then $\\beta_n$, then $\\alpha_n$, and so on, come from collapsing adjacent equalities, which prunes the search tree to a narrow slab around the conjectured bounds.","core_discovery":"The paper's central claim is that three generalizations of the Girard-Newton identities, stated as Identity 3, Identity 4, and Identity 5, form a valid and effective algebraic engine for the PTE, GPTE, and FPTE problems. Identity 3 asserts that a recursively defined sequence $T_k$ attached to $n$ numbers vanishes for every integer $k$, from which the paper derives that certain exponent sets, for example $(h=0,1,2,3,4,5)$ and $(h=-1,0,1,2,3,4)$, have no nontrivial real or integer solutions, and that other systems force structural constraints such as zero sums or zero reciprocal sums. Identity 4 asserts that a family of determinants $W_{nk+r}$ built from power-sum differences satisfies $W_{nm}=\\prod_{i=1}^{n}\\prod_{j=1}^{m}(a_i-b_j)$, with the neighbouring determinants $W_{nk+1}, \\dots, W_{nk+n}$ equal to the elementary symmetric sums of the $a_i$ times $W_{nk}$; the paper reports that this turns the search for ideal solutions into divisibility tests on ratios of determinants, and it recounts finding the smallest known non-symmetric ideal solution of degree 7 this way. Identity 5 gives an analogous determinant calculus for odd exponents, used to locate solutions of types such as $(k=1,3,5,7,9)$ and $(k=1,2,3,5,7,9)$. The paper further claims that ideal solutions of three series of GPTE types carry a constant $C$ equal to the difference of the two associated monic polynomials, and that the normalized GPTE problem satisfies six conjectures determining exact bounds for each variable in descending order; under these conjectures the extreme configurations collapse adjacent variables, and the paper derives polynomial equations for the minimal largest element $\\beta_{n+1,\\min}$ for a range of exponent sets. On its own terms the paper is as much a research programme as a survey: its appendices catalogue 296 types with ideal solutions, many of which it says were located with the methods it introduces.","pith_inferences":["A testable consequence the paper leaves implicit: the same collapse construction that fixes $\\beta_{n+1,\\min}$ should also produce the successive bounds for $\\beta_n$ and $\\alpha_n$, so the polynomials of Example 5.2 and its neighbours for neighbouring exponent sets should be algebraically related; checking how those polynomials factor and relate would map the adjacency structure of the GPTE solut","The trigonometric and algebraic sides of the paper could be welded together: the classical alternating cosine-sum identities behind Identity 12 are standard consequences of root-of-unity spectral arguments, so a proof of Identity 12 along those lines would presumably also settle the small cases of Identities 3–5, which the paper at present supports only by symbolic checks for small parameters.","The constant $C$ viewpoint suggests a reformulation of the search as a problem about integer polynomials of bounded height: an ideal solution exists exactly when two monic polynomials with integer roots differ by a constant polynomial, a finitary condition with a finite search space that would also explain why the observed constants factor into the small primes seen in Chapter 3.","The reported speed-ups, for example thirty minutes to find the first known solution of type $(k=1,2,3,5,7,9)$ after restricting to odd integers, suggest that the bound conjectures carry real computational weight; a clean test would be to rerun a known search without the conjectured bounds and measure how much the search space grows, isolating the contribution of the conjectures from that of the de"],"forward_implications":["Searches for ideal solutions scale to exponent sets beyond the tabulated ones, because the determinant-ratio divisibility test lets a computer discard a candidate tuple on the first non-integer ratio instead of computing and comparing the full power sums.","Existence questions for specific GPTE types become decidable in many cases: the corollaries drawn from Identity 3 already show that types such as $(h=0,1,2,3,4,5)$ and $(h=0,1,4)$ have no nontrivial solutions, and the same method extends the list.","The chain-length trichotomy (no chain, bounded finite chain, or unbounded chain) becomes computable type by type, because the identities bound how many distinct real tuples can share a fixed set of power sums, as the paper demonstrates for types $(h=1,3,4)$ and $(h=2,3,4)$.","The trigonometric identities supply a systematic source of integer solutions whose element ratios approach the sine-squared lower bound $\\beta_{n+1,\\min}=\\sin^2(n\\pi/(2n+2))$ for the PTE case, linking the continuous and discrete sides of the problem.","If the six normalized-GPTE conjectures are correct, then for each exponent set the minimal ideal solution is characterized by the collapsed extreme configuration of Conjecture 2, which would certify the minimality of the appendix solutions and set a target for exhaustive searches."],"supporting_citations":[{"why":"The author's earlier work in which the three generalized Girard-Newton identities were independently derived; it is the source of the paper's central new tools.","marker":"[27]"},{"why":"The classical Girard-Newton identities in the form that Identities 3–5 generalize; it supplies the recursion and symmetric-sum conventions used throughout.","marker":"[112]"},{"why":"Demonstrates that the classical identities solve PTE problems; it is the baseline method that the new generalizations extend and accelerate.","marker":"[19]"},{"why":"Companion demonstration of the classical identities' use in solving PTE; it sets the standard the generalized identities must improve on.","marker":"[20]"},{"why":"The earlier proof of the 6-10-8 identity, which the paper re-derives as a corollary of Identity 4 and uses as an independent check on the new machinery.","marker":"[22]"},{"why":"The standard reference defining ideal solutions and recording the interlacing property for PTE; it is the reference against which the special case of Conjecture 1 is verified.","marker":"[5]"},{"why":"Source of the impossibility result that no nontrivial solutions exist when the number of variables does not exceed the degree; it justifies the ideal-size convention used throughout the paper.","marker":"[60]"},{"why":"The author's earlier systematic study of negative exponents; it supplies the mirror transformation that generates all negative-exponent GPTE types from positive ones.","marker":"[32]"},{"why":"The author's earlier work defining k = 0 as the equal-products case; its convention underlies the GPTE setup and the definition of P_0 in Identity 4.","marker":"[31]"}],"fun_headline_variants":["Three new identities crack the equal-powers puzzle","New identities solve PTE and power-sum equations","PTE problem: three identities, six conjectures","Survey: identities and bounds for PTE searches","Equal-sum search falls to new identity method"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole method stands on the assumption that the determinant identity $W_{nm}=\\prod_{i,j}(a_i-b_j)$ holds with no hidden exceptions for every choice of set sizes $n$ and $m$ and every integer exponent $k$, including the negative and zero powers used to derive the exact bounds in Chapter 5; if that identity fails in some excluded case, the claimed bounds and the fast computer search built on them collapse.","fun_headline_variants_meta":{"raw":{"variants":["Three new identities crack the equal-powers puzzle","New identities solve PTE and power-sum equations","PTE problem: three identities, six conjectures","Survey: identities and bounds for PTE searches","Equal-sum search falls to new identity method"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000287,"raw_usage":{"total_tokens":1876,"prompt_tokens":1329,"completion_tokens":547,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":945,"completion_tokens_details":{"reasoning_tokens":476}},"tokens_in":945,"tokens_out":547,"duration_ms":6902,"temperature":1.0,"reasoning_tokens":476,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:09:08.427391+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The concrete test is a direct spot-check of Identity 4 outside the small cases the paper says it verified symbolically: choose $n=4$, $m=5$, include several negative exponents and the zero-exponent definition of $P_0$, pick random integer tuples with no zero entries and no coincidences $a_i=b_j$, and check numerically whether $W_{20}=\\prod_{i=1}^{4}\\prod_{j=1}^{5}(a_i-b_j)$; one mismatch falsifies the identity as stated, and a pattern of mismatches at zero entries would reveal the missing restriction. A second test targets the search engine itself: recompute the polynomial for $\\beta_{6,\\min}$ of type $(k=1,2,3,4,6)$ exactly as in Example 5.2 by applying Identity 4, and compare its root to a high-precision numerical solution of the collapsed system (5.27); any discrepancy between the two routes refutes the bound conjecture that the search algorithm relies on.","supporting_citations":[{"cited_title":"Two Diophantine systems","cited_arxiv_id":null,"evidence_quote":"The classical Girard-Newton identities in the form that Identities 3–5 generalize; it supplies the recursion and symmetric-sum conventions used throughout."},{"cited_title":"Taylor-series expansion based numerical methods: a primer, performance benchmarking and new approaches for problems with non-smooth solutions","cited_arxiv_id":"2009.11953","evidence_quote":"The earlier proof of the 6-10-8 identity, which the paper re-derives as a corollary of Identity 4 and uses as an independent check on the new machinery."}],"review_version":1}