{"id":"df7ff196-aa64-4c9c-94ad-d3608412ccec","arxiv_id":"1908.01690","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every prime p and every a ≥ 1, the paper gives a closed formula for the p-adic valuation of the Fibonomial coefficient (p^a n choose n)_F in terms of digit sums and the rank z(p).","lead":"Using Fibonacci numbers in place of ordinary integers, this paper derives exact formulas for how many times any prime divides a large family of Fibonomial coefficients. It finishes a divisibility problem that earlier work had only solved for the small primes 2, 3, 5, and 7.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main dependency is the quoted Theorem 2.6 case split; no internal error found, but an independent check of that dependency is warranted.","rationale":"I read the paper as a careful completion of an established program: explicit formulas for ν_p of Fibonomial coefficients, with the main new content being Theorem 3.1 (p=2), Theorem 3.5 (p=5), and Theorem 3.6 (p≠2,5), plus the divisibility corollaries. The proofs are case-based and reference earlier theorems, especially Theorem 2.6 from [14]. I checked representative cases and re-derived the key reduction in Theorem 3.6(iii), Case 2: applying Theorem 2.6 with top exponent a+b and lower exponent b gives exactly the stated contributions to δ, including the b/2, (b+1)/2, and ν_p(F_z(p)) corrections, with A = m−k or m−k−1 according to r>s or r<s. I found no internal inconsistency. The weakest point is structural: the new formulas inherit the correctness of a lengthy quoted case split, and the paper does not display that case analysis in the p|n part of Theorem 3.6(iii). That is the single most load-bearing concern, but it is a verification gap rather than a discovered error. The manuscript itself flags only the deferred p-adic pattern observation, which is not part of the central claim. Since the reader's verdict already identifies this same dependency and the formulas survive my independent checks, I do not see a reason to change the ACCEPT verdict.","tokens_in":15416,"tokens_out":15169,"duration_ms":139036,"concrete_test":"Run an exact-integer brute-force verification of the main theorems: for p ∈ {2,3,5,7,11,13,17,19,23}, a ∈ {1,2,3,4,5}, and all n up to min(500, p^3 · z(p)) covering both p∤n and p|n and all residues r,s, compute ν_p((p^a n choose n)_F) directly from the product formula for Fibonomial coefficients and compare with Theorems 3.1, 3.5, and 3.6. A single mismatch identifies the failing branch of the Theorem 2.6 reduction; if all residues and parities pass, the dependency is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formulas in Theorem 3.6 are proven by reducing to the authors' earlier Theorem 2.6 from [14], a six-case formula with correction terms involving a, ν_p(F_z(p)), ν_p(m_p−k_p), and parity of a. The risk is most concentrated in Theorem 3.6(iii), Case 2 (p ≡ ±2 mod 5, a odd, p | n), where the proof does not display the case analysis: it says the calculation 'is done by the applications of Theorem 2.6' and then asserts the four b-parity/r-s outcomes. If any quoted case in Theorem 2.6(ii) has a misprinted correction term, or if the application mismatches exponents (lower exponent b vs top exponent a+b), the corresponding branch of (3.8), especially the δ term, inherits the error. I re-derived the four subcases from Theorem 2.5(iii) and Theorem 2.4, and the stated δ is consistent: for r>s, b even/odd gives b/2 and (b+1)/2; for r<s gives b/2+ν_p(F_z(p)) and (b−1)/2+ν_p(F_z(p)). So this is a genuine dependency risk, not a detected flaw. The other load-bearing premise, Lemma 2.1(ii) on z(p), is standard and less concerning.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the paper is honest and correct as far as I can tell. It does exactly what the abstract promises: explicit formulas for ν_p((p^a n choose n)_F) for every prime p and every a ≥ 1, plus a clean characterization of when p divides (pn choose n)_F for p ≡ ±2 mod 5. The reader's ACCEPT verdict is sound.\n\nThe genuinely new part is the coverage: previous literature had only p = 2, 3, 5, 7 and a = 1. Here you get Theorems 3.1, 3.5, 3.6, with the nice corollaries about (4n choose n)_F and (8n choose n)_F being odd. The formulas have no free parameters and no invented entities; they depend on digit sums and the rank of appearance. I took representative values such as (16 choose 4)_F and (20 choose 5)_F, and the formulas check out. The proofs are complete case analyses; the appeals to the earlier Theorems 2.4–2.6 are legitimate, since those were proved independently and the assumptions match.\n\nThe soft spots are real but not disqualifying. The main structural one: the new theorems come almost entirely from plugging (p^a n, n) into the earlier general machinery, so the contribution is bookkeeping over a long, quoted case split rather than a new method. The places I would ask a referee to spend time are in the proof of Theorem 3.6(iii), Case 2, where the calculation is asserted as 'done by applications of Theorem 2.6' with the subcases not displayed, and the δ term in (3.8). I re-derived the four subcases and they are consistent, so I don't see an error, but that is the load-bearing spot. Lemma 2.1(ii) is standard. The paper also openly says the pattern in the p-adic representations is deferred to a future paper; that is honest and not a flaw.\n\nCitation pattern is fine: self-citation to [14] is exactly right, since the machinery lives there. No red flags.\n\nThis is for specialists in Fibonacci/Lucas divisibility. It is not a breakthrough, but it is a solid completion of an established program and deserves a serious referee.","headline":"Honest, correct, and genuinely extends coverage to all primes and all a, though the new part is mostly careful bookkeeping on the authors' own earlier theorems.","tokens_in":16340,"tokens_out":2401,"would_cite":true,"duration_ms":23547,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B39","11B65","11A63"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives explicit p-adic valuation formulas for all Fibonomial coefficients (p^a n choose n)_F.","keywords":["Fibonomial coefficients","p-adic valuation","Fibonacci numbers","rank of appearance","digit sum","divisibility","Legendre's formula"],"falsifier":"For p = 3, whose rank of appearance is z(3) = 4, compute nu_3((3n choose n)_F) directly for n = 1 through 50 by factorizing the quotient of Fibonacci products and compare with formula (3.8). For instance, n = 4 gives the predicted value nu_3 = 1 and the direct quotient F_9 F_10 F_11 F_12/(F_1 F_2 F_3 F_4) = 3,994,320 has exactly one factor of 3, while n = 6 gives the predicted value 0; any mismatch in such a finite run would settle the central claim against the paper.","tokens_in":15053,"feed_emoji":"➗","tokens_out":8138,"duration_ms":74666,"temperature":0.7,"pith_summary":"The paper tackles the Fibonomial analogue of the binomial coefficient, where factorials are replaced by products of consecutive Fibonacci numbers. Earlier work could decide when a prime divides these numbers only for p = 2, 3, 5, 7 and for the exponent a = 1. Here the authors claim a complete explicit formula for the exponent of any prime p in (p^a n choose n)_F, for every positive a and n. The payoff is a clean decision rule: divisibility of (p n choose n)_F by p is controlled by a base-p digit sum and a residue modulo the rank of appearance z(p).","feed_headline":"Closed formulas give p-adic valuations of Fibonomial coefficients","feed_subtitle":"For every prime p and all a, n, the formulas decide when p divides (p^a n choose n)_F.","key_machinery":"The rank of appearance z(p), the smallest positive integer with p | F_{z(p)}, plays the role that the modulus p plays for ordinary binomial coefficients. The formulas reduce Fibonomial valuations to ordinary binomial valuations on the integer A = floor(n(p^a-1)/($p^{{nu_p(n)}}$ z(p))) and then add correction terms indexed by residues modulo z(p) and by nu_p(F_{z(p)}). The digit-sum identities come from Legendre's formula applied to A!.","core_discovery":"For every prime p and positive integers a and n, the p-adic valuation of the Fibonomial coefficient (p^a n choose n)_F is given by one of three closed formulas. For p = 2, Theorem 3.1 expresses nu_2((2^a n choose n)_F) as delta + s_2(A) minus an indicator term, where A = floor((2^a-1)n/(3 * $2^{{nu_2(n)}}$)) and delta, epsilon are small residue indicators. For p = 5, Theorem 3.5 gives nu_5((5^a n choose n)_F) = s_5((5^a-1)n)/4. For p not equal to 2 or 5, Theorem 3.6 writes nu_p as a difference involving A = floor(n(p^a-1)/($p^{{nu_p(n)}}$ z(p))), the base-p digit sum s_p(A), and the residues r = p^a n mod z(p), s = n mod z(p), with an extra correction delta when p is congruent to pm 2 mod 5 and a is odd. The corollaries then characterize when p divides (p n choose n)_F: for p congruent to pm 1 mod 5 the criterion is s_p(A) >= p-1, while for p congruent to pm 2 mod 5 it is a threshold on s_p(A) together with residue conditions.","pith_inferences":["The threshold conditions s_p(A) >= c, combined with residue conditions modulo z(p), define sets of n that a finite automaton reading base-p digits can recognize, so the divisibility pattern is effectively automatic in base p; the paper notes a pattern but does not formalize this.","The same reduction may extend to other nondegenerate Lucas sequences U by replacing z(p) with the rank of appearance of p in U whenever the analogue of Lemma 2.1(ii) holds; the paper works only with Fibonacci numbers.","The appearance of s_2(A) for p = 2 suggests a Kummer-style carry interpretation: the 2-adic valuation of a Fibonomial coefficient counts carries in a Fibonacci-based numeral system, parallel to Kummer's theorem for ordinary binomial coefficients."],"forward_implications":["When p is congruent to pm 1 mod 5, Corollary 3.9 gives the exact criterion p | (p n choose n)_F if and only if s_p(A) >= p-1, with A = n(p-1)/(p^{nu_p(n)} z(p)); this is a base-p digit-sum test.","When p is congruent to pm 2 mod 5, Corollary 3.8 shows divisibility is governed by s_p(A) compared with (a/2)(p-1) or ((a+1)/2)(p-1), with automatic divisibility in the odd-exponent cases when r differs from s and p divides n, or when r < s and p does not divide n.","For p = 2, Corollaries 3.3 and 3.4 give that (4n choose n)_F is odd exactly for powers of two, and (8n choose n)_F is odd exactly for n = (1 + 3*2^k)/7 with k congruent to 1 mod 3.","For p = 5, every (5^a n choose n)_F is divisible by 5, and its exact 5-adic valuation is s_5((5^a-1)n)/4.","For p = 2 and a = 1, the formula recovers the known result that (2n choose n)_F is even for all n >= 2."],"supporting_citations":[{"why":"Supplies Theorems 2.4, 2.5, and 2.6, the valuation lemmas the new proofs invoke for every case split.","marker":"[14]"},{"why":"Gives the Kummer-like divisibility rule for Lucasnomials that previously extended the known p = 2, 3 and p = 5, 7 results, and forms the line of work this paper generalizes.","marker":"[1]"},{"why":"Provides the power-of-prime rule for generalized binomial coefficients on which the Kummer-like machinery rests.","marker":"[8]"},{"why":"Previously determined when (2n choose n)_F is even; that result is recovered here as Corollary 3.2.","marker":"[9]"},{"why":"Previously handled divisibility of Fibonomial coefficients by 3, one of the special cases the all-p formulas generalize.","marker":"[10]"}],"fun_headline_variants":["All primes covered: explicit p-adic valuations of Fibonomial coefficients","Deciding p-divisibility of Fibonomial coefficients via closed formulas","Explicit valuation formulas for all prime powers in Fibonomial coefficients","Closed forms for p-adic valuations of Fibonomial coefficients, any p","When does p divide Fibonomial coefficients? Closed formulas decide"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The new formulas inherit every case condition and correction term from the authors' earlier valuation lemmas, especially the parity and residue split in Theorem 2.6, so if any one of those inherited terms is wrong, the matching case of the new formula is wrong; the argument also assumes z(p) divides p+1 when p is congruent to pm 2 mod 5 and divides p-1 otherwise.","fun_headline_variants_meta":{"raw":{"variants":["All primes covered: explicit p-adic valuations of Fibonomial coefficients","Deciding p-divisibility of Fibonomial coefficients via closed formulas","Explicit valuation formulas for all prime powers in Fibonomial coefficients","Closed forms for p-adic valuations of Fibonomial coefficients, any p","When does p divide Fibonomial coefficients? Closed formulas decide"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000809,"raw_usage":{"total_tokens":3559,"prompt_tokens":962,"completion_tokens":2597,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":2503}},"tokens_in":578,"tokens_out":2597,"duration_ms":18594,"temperature":1.0,"reasoning_tokens":2503,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:08:25.740894+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For p = 3, whose rank of appearance is z(3) = 4, compute nu_3((3n choose n)_F) directly for n = 1 through 50 by factorizing the quotient of Fibonacci products and compare with formula (3.8). For instance, n = 4 gives the predicted value nu_3 = 1 and the direct quotient F_9 F_10 F_11 F_12/(F_1 F_2 F_3 F_4) = 3,994,320 has exactly one factor of 3, while n = 6 gives the predicted value 0; any mismatch in such a finite run would settle the central claim against the paper.","supporting_citations":[{"cited_title":"Phunphayap and P","cited_arxiv_id":null,"evidence_quote":"Supplies Theorems 2.4, 2.5, and 2.6, the valuation lemmas the new proofs invoke for every case split."},{"cited_title":"Ballot, Divisibility of Fibonomials and Lucasnomial s via a general Kummer rule, Fibonacci Quart","cited_arxiv_id":null,"evidence_quote":"Gives the Kummer-like divisibility rule for Lucasnomials that previously extended the known p = 2, 3 and p = 5, 7 results, and forms the line of work this paper generalizes."},{"cited_title":"Knuth and H","cited_arxiv_id":null,"evidence_quote":"Provides the power-of-prime rule for generalized binomial coefficients on which the Kummer-like machinery rests."},{"cited_title":"Marques and P","cited_arxiv_id":null,"evidence_quote":"Previously determined when (2n choose n)_F is even; that result is recovered here as Corollary 3.2."},{"cited_title":"Marques and P","cited_arxiv_id":null,"evidence_quote":"Previously handled divisibility of Fibonomial coefficients by 3, one of the special cases the all-p formulas generalize."}],"review_version":1}