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A10 Integers 13 (2013) on Divisibility Properties Of #A10 INTEGERS 13 (2013) ON DIVISIBILITY PROPERTIES OF SOME DIFFERENCES OF THE CENTRAL BINOMIAL COEFFICIENTS AND CATALAN NUMBERS Tam´as Lengyel Department of Mathematics, Occidental College, Los Angeles, California [email protected] Received: 5/1/12, Revised: 10/16/12, Accepted: 2/24/13, Published: 3/15/13 Abstract We discuss divisibility properties of some differences of the central binomial coeffi- cients and Catalan numbers. The main tool is the application of various congruences modulo high prime powers for binomial coefficients combined with some recurrences relevant to these combinatorial quantities. 1. Introduction The differences of certain combinatorial quantities exhibit interesting divisibility properties. For example, we can consider differences of central binomial coefficients 2n c = , n 0, n n ≥ and Catalan numbers 1 2n C = , n 0. n n + 1 n ≥ We will need some basic notation. Let n and k be positive integers, p be a prime, dp(k) and νp(k) denote the sum of digits in the base p representation of k and the highest power of p dividing k, respectively. The latter one is often referred to as the p-adic order of k. For the rational n/k we set ν (n/k) = ν (n) ν (k). p p − p Our main results concern the p-adic order of the differences of central bino- mial coefficients c n+1 c n (cf. Theorems 2.1 and 2.4), Catalan numbers ap +b − ap +b C n+1 C n (cf. Theorems 2.2 and 2.11, and Remark 4.1) with a prime p, ap +b − ap +b (a, p) = 1, and n n for some integer n 0. These results are essential in ≥ 0 0 ≥ obtaining the p-adic order of the differences of certain Motzkin numbers, more pre- cisely M n+1 M n with a prime p and different settings of a and b that are ap +b − ap +b discussed in [8] and [9]. Of course, results involving the exact orders of differences INTEGERS 13 (2013) 2 or lower bounds on them can be easily rephrased in terms of super congruences for the underlying quantities. Section 2 collects some of the main results while Section 3 is devoted to some known results and their direct consequences regarding congruential and p-adic prop- erties of the binomial coefficients, e.g., Corollary 3.9. Section 4 contains the proofs of the main results of Section 2 and presents some interesting congruences for cen- tral binomial coefficients (cf. Corollary 4.1 improving Corollary 3.5) and Catalan numbers (cf. Theorem 2.12). In Section 5, we derive and prove some important p-adic properties of the differences of certain harmonic numbers stated in Section 2. These properties are important in the actual use of Theorem 2.4. 2. Main Results We state some of the main results of the paper. Theorem 2.1. For p = 2 and a odd, we have that 2a ν (c n+1 c n ) = 3(n + 1) + ν 1 = 3(n + 1) + d (a) 1, n 1. 2 a2 − a2 2 a − 2 − ≥ For p = 3 and (a, 3) = 1, 2a ν (c n+1 c n ) = 3(n + 1) + ν 1, n 0. 3 a3 − a3 3 a − ≥ Let B denote the nth Bernoulli number. For any prime p 5, (a, p) = 1, and n ≥ νp(Bp 3) = 0 or 1, we have that − − 2a νp(capn+1 capn ) = 3(n + 1) + νp + νp(Bp 3), n 0. − a − ≥ 2a Note. The term νp( a ) contributes to the p-adic orders above exactly if at least one of the p-ary digits of a is at least as large as p/2. Remark 2.1. It is well known that ν (B ) 1 by the von Staudt–Clausen the- p n ≥ − orem. If the prime p divides the numerator of Bp 3, i.e., νp(Bp 3) 1, or equiv- − − ≥ alently 2p 2 mod p4, then it is sometimes called a Wolstenholme prime [2]. p ≡ The only known Wolstenholme primes up to 109 are p = 16843 and 2124679. For such primes Theorem 2.1 is inconclusive and gives only the lower bound 2a 3(n + 1) + νp a + 1 on the p-adic order. For the p-adic order of the differences of Catalan numbers we obtain the following theorem. INTEGERS 13 (2013) 3 Theorem 2.2. For any prime p 2 and (a, p) = 1, we have ≥ 2a ν (C n+1 C n ) = n + ν , n 1. p ap − ap p a ≥ The case with n = 0 is slightly different and included in Lemma 2.3. For n 1 and any prime p 2, we have ν (C n ) = ν (c n ) = ≥ ≥ p ap p ap ν 2a and ν (C C ) = ν (C ) if in addition (a, p) = 1. p a p ap − a p a The nature of the p-adic order in Theorem 2.1 changes as we introduce an additive term b, and we get a result similar to Theorem 2.2. Theorem 2.4. For p = 2, a odd, and n n = 1 we have ≥ 0 2a ν (c n+1 c n ) = n + ν + 1, 2 a2 +1 − a2 +1 2 a and in general, for b 1 and n n = log 2b ≥ ≥ 0 2 2a ν (c n+1 c n ) = n + ν + ν (f(b)) 2 a2 +b − a2 +b 2 a 2 = n + d (a) + d (b) log b , 2 2 − 2 where f(b) = 2 2b (H H ) with H = n 1/j being the nth harmonic number. b 2b − b n j=1 For any prime p 3, (a, p) = 1, and b 1 we have that ≥ ≥ 2a ν (c n+1 c n ) = n + ν + ν (f(b)), p ap +b − ap +b p a p for n n = max ν (f(b)) + 2r ν 2b + 1, r + 1 = max ν (2(H H )) + ≥ 0 { p − p b } { p 2b − b 2r + 1, r + 1 and r = log 2b . } p In general, for any prime p 3, (a, p) = 1, b 1, and n > log 2b , we have ≥ ≥ p 2a 2b ν (c n+1 c n ) n + ν + ν log 2b p ap +b − ap +b ≥ p a p b − p and for p = 2, a odd, and n log 2b ≥ 2 2a 2b ν (c n+1 c n ) n + ν + ν log b . 2 a2 +b − a2 +b ≥ 2 a 2 b − 2 We can make some useful statements on the magnitude of νp(f(b)). By taking the common denominator in H H , we note that ν (f(b)) ν 2b log 2b 2b − b p ≥ p b − p for p 3 and ν (f(b)) = d (b) log b . This implies that ν (f(b)) 0 for ≥ 2 2 − 2 p ≥ 1 b (p 1)/2. In this range the binomial factor of f(b) can be dropped, leaving ≤ ≤ − only ν (H H ). It appears that ν (f(b)) = 0 in many cases, however f(2) = 7 p 2b − b p and f(15) = 450351518582/2145; thus, νp(f(2)) = 1 and νp(f(15)) = 2 with p = 7. Also note that for p 3 we have ν (f(b)) = 0 if b + 1 p 2b and ν (f(p)) = 1. ≥ p ≤ ≤ p − In fact, a much stronger statement about νp(f(b)) is given in INTEGERS 13 (2013) 4 Theorem 2.5. For p = 2 and c 1, we have ν (f(c)) = d (c) log c . For ≥ 2 2 − 2 p 3, we have ν (f(pk)) = k, k 1, and in general, for c 1 and k 0, we have ≥ p − ≥ ≥ ≥ ν (f(cpk)) = k + ν (f(c)) provided that ν (H H ) 0. p − p p 2c − c ≤ We observe that νp(f(b)); and therefore, the p-adic order of the difference c n+1 c n changes only slightly for bs that are small relative to p. ap +b − ap +b Clearly, ν (f(b)) 1 and equality holds only if b+1 is a power of two. We observe 2 ≤ that ν (f(b)) 0 and equality holds exactly if b = 1 or all ternary digits of b are 3 ≤ at least one and b 2 mod 3, i.e., the least significant digit is two, as these facts ≡ follow from Theorem 2.8. Clearly, the study of νp(f(b)) requires the understanding of the behavior of ν (H H ). The next lemma and theorem are straightforward. p 2b − b Lemma 2.6. For any prime p and integer b 1, we have ν (H H ) ≥ p 2b − b ≥ log 2b . − p This follows immediately as the exponent in the largest power of p can not exceed log 2b . Also note the following p Theorem 2.7. For the positive integers b < a, H H is never an integer. a − b We include the standard proof of this theorem which, with some tweaking, leads to the proof of the following theorem. Indeed, for the exact orders, we get Theorem 2.8. For p = 2 and 3, we have ν (H H ) = log 2b . For p = 5 p 2b − b − p we have ν5(H2b Hb) = log5 2b + χ m: 3 5m<b<2 5m (2.1) − − ∃ 2 · · with the indicator variable χ m: 3 5m<b<2 5m which is 1 if for b there exists an m so ∃ 2 · · that 3 5m < b < 2 5m and otherwise, it is 0. 2 · · For primes larger than 5 it seems more complicated to establish the exact order of ν (H H ).
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