Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients II

Phakhinkon Phunphayap; Prapanpong Pongsriiam

Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients II

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Pith's one-line read This paper derives explicit p-adic valuation formulas for all Fibonomial coefficients (p^a n choose n)_F.

desk verdict 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. read the letter →

arxiv 1908.01690 v1 pith:J6SXP5Q5 submitted 2019-08-05 math.NT

Phakhinkon Phunphayap , Prapanpong Pongsriiam This is my paper · ORCID

classification math.NT MSC 11B3911B6511A63

keywords Fibonomialcoefficientsp-adicvaluationFibonaccinumbersrankofappearancedigitsumdivisibilityLegendre'sformula

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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).

What carries the argument

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!.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

Citation pattern is fine: self-citation to [14] is exactly right, since the machinery lives there. No red flags.

This 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main formulas are specializations of the authors' independently established prior theorems, not re-statements of the target results.

full rationale

The derivation chain is a sequence of reductions to earlier general results, chiefly Theorems 2.4, 2.5, and 2.6 from the authors' previous paper [14]. This is self-citation, but it is not circular. The quoted theorems in [14] are parameter-free general formulas for p-adic valuations of Fibonomial coefficients (for arbitrary arguments and exponents), and their stated assumptions do not include the target formulas of the present paper. The current paper obtains (3.1), (3.2), (3.6), (3.7), and (3.8) by substituting n = p^b * ell or m = floor(p^a n / z(p)), k = floor(n / z(p)), and then applying Legendre's formula and elementary floor manipulations. No fitted parameter is renamed as a prediction, and no target formula is assumed in the input. The rank of appearance facts in Lemma 2.1 are standard and are cited from [14, Lemma 1] as well-known results, not as a uniqueness or forcing argument. The concentration of the proof of Theorem 3.6(iii), Case 2 into 'the applications of Theorem 2.6' creates a genuine dependency risk: if a case split or additive correction in Theorem 2.6 were erroneous, the corresponding branch of (3.8) would inherit the error. However, dependency on a previously proved, separately refereed theorem is not circularity. The reduction is explicit and checkable, and no step in the paper defines its conclusion into its hypotheses.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters: the formulas involve only a, n, p, z(p), digit sums and fractional parts, all standard or defined in Section 2. No invented entities: the paper introduces no new objects, forces, or constants. The axioms are either classical (Legendre's formula, rank-of-apparition facts) or the authors' own published theorems from [14], which are parameter-free and do not assume the target results. The main external burden is trust in the correctness of the quoted case analyses in [14].

assumptions (5)

domain assumption Lemma 2.1: n | F_m iff z(n) | m; z(p) | p+1 iff p ≡ ±2 (mod 5), otherwise z(p) | p−1; gcd(z(p),p) = 1.
Rank-of-apparition facts cited to the authors' own [14, Lemma 1] rather than proved. They determine which branch of Theorem 3.6 applies and force p^a ≡ ±1 (mod z(p)), the engine of the case analysis.
domain assumption Theorem 2.4 (from [14, Theorem 7]): closed form for ν_p(⌊ℓ p^a / m⌋!).
Quoted without proof; converts factorial valuations inside the main calculations into explicit algebraic expressions. Parameter-free, assumptions stated (p ≡ ±1 (mod m)), and does not assume the target formulas.
domain assumption Theorem 2.5 (from [14, Theorem 11 and Corollary 12]): Kummer-type formulas for ν_2 and ν_5 of Fibonomials, and ν_p((m choose k)_F) = ν_p((⌊m/z(p)⌋ choose ⌊k/z(p)⌋)) plus a correction when m mod z(p) < k mod z(p).
The bridge from Fibonomial valuations to ordinary binomial valuations; backbone of Theorem 3.6 and Theorem 3.5. Quoted without proof.
domain assumption Theorem 2.6 (from [14, Theorem 13]): the case-split formula for ν_p((ℓ_1 p^b choose ℓ_2 p^a)_F) by parities of a and b and residues modulo z(p).
The most fragile input: Theorem 3.6(iii) and portions of Theorem 3.1 inherit this formula verbatim, including parity-dependent ⌈a/2⌉, ⌈b/2⌉ terms and the ν_p(F_{z(p)}) corrections.
standard math Legendre's formula ν_p(n!) = (n − s_p(n))/(p − 1), and s_q(2^c m) = s_q(m) for integers c, m.
Used throughout to translate factorial valuations into digit sums, e.g., the second equalities in (3.1), (3.2), (3.6) and the proofs of Corollaries 3.3 and 3.8.

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Cite this review

Pith. "Pith review of Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients II." pith.science (2026). https://pith.science/paper/J6SXP5Q5

@misc{pith2026190801690,
  author       = {Pith},
  title        = {Pith review of: Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6SXP5Q5}},
  note         = {Machine review of arXiv:1908.01690}
}

abstract

In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that ${pn \choose n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [14]

Phunphayap and P

P. Phunphayap and P. Pongsriiam, Explicit formulas for the p-adic valuations of Fibonomial coef- ﬁcients, J. Integer Seq. 21 (2018), Article 18.3.1

work page 2018

[2] [1]

Ballot, Divisibility of Fibonomials and Lucasnomial s via a general Kummer rule, Fibonacci Quart

C. Ballot, Divisibility of Fibonomials and Lucasnomial s via a general Kummer rule, Fibonacci Quart. 53 (2015), 194–205

work page 2015

[3] [2]

Ballot, The congruence of Wolstenholme for generaliz ed binomial coeﬃcients related to Lucas sequences, J

C. Ballot, The congruence of Wolstenholme for generaliz ed binomial coeﬃcients related to Lucas sequences, J. Integer Seq. 18 (2015), Article 15.5.4

work page 2015

[4] [3]

Ballot, Lucasnomial Fuss-Catalan numbers and relate d divisibility questions, J

C. Ballot, Lucasnomial Fuss-Catalan numbers and relate d divisibility questions, J. Integer Seq. 21 (2018), Article 18.6.5

work page 2018

[5] [4]

Ballot, Divisibility of the middle Lucasnomial coeﬃc ient, Fibonacci Quart

C. Ballot, Divisibility of the middle Lucasnomial coeﬃc ient, Fibonacci Quart. 55 (2017), 297–308

work page 2017

[6] [5]

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Math ematics : A Foundation for Com- puter Science, Second Edition, Addison–Wesley, 1994

work page 1994

[7] [6]

Khaochim and P

N. Khaochim and P. Pongsriiam, The general case on the ord er of appearance of product of consecutive Lucas numbers, Acta Math. Univ. Comenian. 87 (2 018), 277–289

work page

[8] [7]

Khaochim and P

N. Khaochim and P. Pongsriiam, On the order of appearance of product of Fibonacci numbers, Contrib. Discrete Math. 13 (2018), 45–62

work page 2018