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In this work, the p-adic valuation of Eulerian numbers is explored. A tree whose nodes contain information about the p-adic valuation of these numbers is constructed, and this tree, along with some classical results for Bernoulli numbers, is used to compute the exact p divisibility for the Eulerian numbers when the first variable lies in a congruence class and p satisfies some regularity properties.

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... For some identities involving Fibonomial coefficients and generalizations, we refer the reader to the work of Kilic and his coauthors [7,8,[18][19][20][21]. For the p-adic valuations of Eulerian, Bernoulli, and Stirling numbers, see [6,9,14,23,40]. Hence the relation p | p a n n F has been studied only in the case p = 2, 3, 5, 7 and a = 1. ...
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In this article, we give explicit formulas for the p-adic valuations of the Fibonomial coefficients (pann)F\binom{p^a n}{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=2,3,5,7p = 2,3,5,7 and a=1a = 1. Then we use these formulas to characterize the positive integers n such that (pnn)F\binom{pn}{n}_F is divisible by p, where p is any prime which is congruent to ±2(mod5)\pm 2 \pmod{5}.
... Valuations have been studied for the Stirling numbers S(n, k) [2,6], sequences satisfying first-order recurrences [3], the Fibonacci numbers [17], the ASM (alternating sign matrices) numbers [7,20], and coefficients of a polynomial connected to a quartic integral [1,8,21]. Other results of this type appear in [4,[11][12][13]19]. Consider the sequence of valuations . ...
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For a prime p and an integer x, the p-adic valuation of x is denoted by . For a polynomial Q with integer coefficients, the sequence of valuations is shown to be either periodic or unbounded. The first case corresponds to the situation where Q has no roots in the ring of p-adic integers. In the periodic situation, the period length is determined.
... Examples of such sequences include the Stirling numbers S(n, k) [3,6], sequences satisfying first order recurrences [4], the Fibonacci numbers [17], the ASM (alternating sign matrices) numbers [7,21], coefficients of a polynomial connected to a quartic integral [2,8,18,22]. Other results of this type appear in [1,11,12,13,20]. ...
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For a prime p and an integer x, the p-adic valuation of x is denoted by νp(x)\nu_{p}(x). For a polynomial Q with integer coefficients, the sequence of valuations νp(Q(n))\nu_{p}(Q(n)) is shown to be either periodic or unbounded. The first case corresponds to the situation where Q has no roots in the ring of p-adic integers. In the periodic situation, the exact period is determined.
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The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties.
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this paper p (r) denotes the exponent of the highest power of a prime p which divides r and is referred to as the p-adic order of r. We characterize the p-adic orders p (Fn ) and p (Ln ), i.e., the exponents of a prime p in the prime power decomposition of Fn and Ln , respectively. The characterization of the divisibility properties of combinatorial quantities has always been a popular area of research. In particular, finding the highest powers of primes which divide these numbers (e.g., factorials, binomial coefficients [14], Stirling numbers ([2], [1], [10], and [9]) has attracted considerable attention. The analysis of the periodicity modulo any integer ([3], [11], [14], and [8]) of these numbers helps exploring their divisibility properties (e.g., [9]). The periodic property of the Fibonacci and Lucas numbers has been extensively studied (e.g., [16], [13], [17], and [12]). Here we use some of these properties and methods to find p (Fn ) and p (Ln ): An application of the results to the Stirling numbers of the second kind is discussed at the end of the paper. We note that Halton [5] obtained similar results on the p-adic order of the Fibonacci numbers, and additional references on earlier developments can be found in Robinson [13] and Vinson [15]. The approach presented here is based on a refined analysis of the periodic structure of the Fibonacci numbers by exploring its properties, in particular, around the points where Fn j 0 (mod p): (The smallest n such that Fn j 0 (mod p) is called the rank of apparition of prime p and is denoted by n(p).) This technique is based on that of Wilcox [17] and provides a simple and self-contained analysis of properties related to divisibility. For instance, we obtain another characterization of the ratio of the period to the...
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Lecture I The Early History of Fermat's Last Theorem.- 1 The Problem.- 2 Early Attempts.- 3 Kummer's Monumental Theorem.- 4 Regular Primes.- 5 Kummer's Work on Irregular Prime Exponents.- 6 Other Relevant Results.- 7 The Golden Medal and the Wolfskehl Prize.- Lecture II Recent Results.- 1 Stating the Results.- 2 Explanations.- Lecture III B.K. = Before Kummer.- 1 The Pythagorean Equation.- 2 The Biquadratic Equation.- 3 The Cubic Equation.- 4 The Quintic Equation.- 5 Fermat's Equation of Degree Seven.- Lecture IV The Naive Approach.- 1 The Relations of Barlow and Abel.- 2 Sophie Germain.- 3 Congruences.- 4 Wendt's Theorem.- 5 Abel's Conjecture.- 6 Fermat's Equation with Even Exponent.- 7 Odds and Ends.- Lecture V Kummer's Monument.- 1 A Justification of Kummer's Method.- 2 Basic Facts about the Arithmetic of Cyclotomic Fields.- 3 Kummer's Main Theorem.- Lecture VI Regular Primes.- 1 The Class Number of Cyclotomic Fields.- 2 Bernoulli Numbers and Kummer's Regularity Criterion.- 3 Various Arithmetic Properties of Bernoulli Numbers.- 4 The Abundance of Irregular Primes.- 5 Computation of Irregular Primes.- Lecture VII Kummer Exits.- 1 The Periods of the Cyclotomic Equation.- 2 The Jacobi Cyclotomic Function.- 3 On the Generation of the Class Group of the Cyclotomic Field.- 4 Kummer's Congruences.- 5 Kummer's Theorem for a Class of Irregular Primes.- 6 Computations of the Class Number.- Lecture VIII After Kummer, a New Light.- 1 The Congruences of Mirimanoff.- 2 The Theorem of Krasner.- 3 The Theorems of Wieferich and Mirimanoff.- 4 Fermat's Theorem and the Mersenne Primes.- 5 Summation Criteria.- 6 Fermat Quotient Criteria.- Lecture IX The Power of Class Field Theory.- 1 The Power Residue Symbol.- 2 Kummer Extensions.- 3 The Main Theorems of Furtwangler.- 4 The Method of Singular Integers.- 5 Hasse.- 6 The p-Rank of the Class Group of the Cyclotomic Field.- 7 Criteria of p-Divisibility of the Class Number.- 8 Properly and Improperly Irregular Cyclotomic Fields.- Lecture X Fresh Efforts.- 1 Fermat's Last Theorem Is True for Every Prime Exponent Less Than 125000.- 2 Euler Numbers and Fermat's Theorem.- 3 The First Case Is True for Infinitely Many Pairwise Relatively Prime Exponents.- 4 Connections between Elliptic Curves and Fermat's Theorem.- 5 Iwasawa's Theory.- 6 The Fermat Function Field.- 7 Mordell's Conjecture.- 8 The Logicians.- Lecture XI Estimates.- 1 Elementary (and Not So Elementary) Estimates.- 2 Estimates Based on the Criteria Involving Fermat Quotients.- 3 Thue, Roth, Siegel and Baker.- 4 Applications of the New Methods.- Lecture XII Fermat's Congruence.- 1 Fermat's Theorem over Prime Fields.- 2 The Local Fermat's Theorem.- 3 The Problem Modulo a Prime-Power.- Lecture XIII Variations and Fugue on a Theme.- 1 Variation I (In the Tone of Polynomial Functions).- 2 Variation II (In the Tone of Entire Functions).- 3 Variation III (In the Theta Tone).- 4 Variation IV (In the Tone of Differential Equations).- 5 Variation V (Giocoso).- 6 Variation VI (In the Negative Tone).- 7 Variation VII (In the Ordinal Tone).- 8 Variation VIII (In a Nonassociative Tone).- 9 Variation IX (In the Matrix Tone).- 10 Fugue (In the Quadratic Tone).- Epilogue.- Index of Names.
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We analyze properties of the 2-adic valuations of S(n,k), the Stirling numbers of the second kind. For fixed kNk \in \mathbb{N}, a conjectured pattern for the valuation is provided in terms of the dyadic format of n. This conjecture is established for k=5.
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Let n,k,a and c be positive integers and b be a nonnegative integer. Let v2(k)v_2(k) and s2(k)s_2(k) be the 2-adic valuation of k and the sum of binary digits of k, respectively. Let S(n,k) be the Stirling number of the second kind. We first show that v2(S(c2n,b2n+1+a))s2(a)1,v_2(S(c2^n,b2^{n+1}+a))\geq s_2(a)-1, where 0<a<2n+10<a<2^{n+1} and 2c2\nmid c. Further, we prove that v2(S(c2n,(c1)2n+a))=s2(a)1v_2(S(c2^{n},(c-1)2^{n}+a))=s_2(a)-1, where n2n\geq 2, 1a2n1\leq a\leq 2^n and 2c2\nmid c. Finally, we show that if 3k2n3\leq k\leq 2^n and k is not a power 2 minus 1, then v2(S(a2n,k)S(b2n,k))=n+v2(ab)log2k+s2(k)+δ(k),v_2(S(a2^{n},k)-S(b2^{n},k)) =n+v_2(a-b)-\lceil\log_2k\rceil +s_2(k)+\delta(k), where δ(4)=2\delta(4)=2, δ(k)=1\delta(k)=1 if k>4k>4 is a power of 2, and δ(k)=0\delta(k)=0 otherwise. This confirms a conjecture of Lengyel raised in 2009 except that k is a power of 2 minus 1.
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A prime p is called a Fibonacci-Wieferich prime if F-p-(p/5) 0 ( mod p(2)), where F-n is the nth Fibonacci number. We report that there exist no such primes p < 2 x 10(14). A prime p is called a Wolstenholme prime if ((2p-1)(p-1)) 1 (mod p(4)). To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes p < 10(9). Wolstenholme, in 1862, proved that ((2p-1)(p-1)) (mod p(3)) for all primes p >= 5. It is estimated by a heuristic argument that the "probability" that p is Fibonacci-Wieferich ( independently: that p is Wolstenholme) is about 1/p. We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient Fp-(p/5)/p modulo p.