Proof of a conjecture of Amdeberhan and Moll on a divisibility property of binomial coefficients

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1 Proof of a cojecture of Amdeberha ad Moll o a divisibility property of biomial coefficiets Qua-Hui Yag School of Mathematics ad Statistics Najig Uiversity of Iformatio Sciece ad Techology Najig, PR Chia yagquahui01@163com Submitted: Feb 10, 2014; Accepted: Dec 21, 2014; Published: Ja 9, 2015 Mathematics Subject Classificatios: 05A10, 11B65 Abstract Let a, b ad be positive itegers with a > b I this ote, we prove that ( ( ( 2 2a a (2 + 1( (a b(3a b a This cofirms a recet cojecture of Amdeberha ad Moll Keywords: biomial coefficiets; p-adic order; divisibility properties 1 Itroductio I 2009, Bober [1] determied all cases such that (a 1! (a k! (b 1! (b k+1! Z, where a s b t for all s, t, a s = b t ad gcd(a 1,, a k, b 1,, b k+1 = 1 Recetly, Z-W Su [12, 13] studied divisibility properties of biomial coefficiets ad obtaied some iterestig results For example, ( ( ( (2 + 1, 3 Supported by the Natioal Natural Sciece Foudatio of Chia, Grat No Startup Foudatio for Itroducig Talet of NUIST, Grat No 2014r ad the the electroic joural of combiatorics 22(1 (2015, #P19 1

2 ( 3 ( ( 15 5 ( Later, Guo ad Krattethaler (see [5, 8] obtaied some similar divisibility results Related results appear i [2]-[4] ad [7]-[11] Itroduce the otatio S = ( 6 3 3( 2(2 + 1 ( 2 ad t = ( 15 5 ( 51 1 ( ( 3 I [6], Guo proved the cojectures due to ZW Su [12, 13] Theorem A ([12, Cojecture 3(i] Let be a positive iteger The 3S 0 (mod Theorem B ([13, Cojecture 13] Let be a positive iteger The 21t 0 (mod Recetly, T Amdeberha ad V H Moll proposed a cojecture related to Theorems A ad B, which was oly preseted as Cojecture 71 i Guo s paper [6] by private commuicatio This otes provides a proof of this cojecture Theorem 1 Let a, b ad be positive itegers with a > b The ( ( 2 2a (2 + 1( (a b(3a b a ( a Remark 2 Theorem A is the special case a = 3, b = 1 of Theorem 1 2 Proofs For a real umber z, deote the greatest iteger ot exceedig z by z ad {z} deotes the fractioal part of z For a iteger ad a prime p, write p k if p k ad p k+1 The iteger k above is deoted by ν p ( It is well kow that ν p (! = (1 The proof of Theorem 1 begis with a prelimiary result i=1 Lemma 3 Let x ad y be two real umbers The 2x + y x + x y + 2y the electroic joural of combiatorics 22(1 (2015, #P19 2

3 Proof The idetity 2x + y = x + (x y + 2y shows that it suffices to {2x} + {y} {x} + {x y} + {2y} The proof ow follows by comparig {x} ad {y} to 1/2 The details are left to the reader Proof of Theorem 1 Let T (a, b, := ( 2a a ( a /( 2 = (2a!(! (a!(a!(2! By (1, for ay prime p, ν p (T (a, b, = ( 2a a a 2 i=1 Lemma 3 shows that each term of ν p (T (a, b, is oegative Hece ν p (T (a, b, 0 Therefore, T (a, b, Z Sice gcd(2 + 1, = 1, it suffices to prove that ad (a b(3a bt (a, b, (a b(3a bt (a, b, The secod statemet is established here The proof of the first statemet is similar ad the details are omitted Suppose that p α with α 1 It is show that p α 3(a b(3a bt (a, b, (2 Let p β a b ad p γ 3a b with β 0 ad γ 0 Write τ = max{β, γ} If α τ, the (2 clearly holds Now we assume α > τ Suppose that p 5 The statemet 2a a a 2 = 1 is established for i = τ + 1, τ + 2,, α This is prove ext Notig that p ad p 5, it follows that gcd(p, = 1 Observe that p α 2 + 3, it follows that 2 p α 3 (mod p α ad (p α 3/2 (mod p α Take i {τ + 1, τ + 2,, α} The 2 3 (mod ad ( 3/2 (mod Now we divide ito several cases accordig to the value of a (mod Case 1 a t (mod with 0 t < ( 3/2 It follows that 2a 2t (mod ad 0 2t < 3 Also a t ( 3/2 + (mod, the electroic joural of combiatorics 22(1 (2015, #P19 3

4 where 0 t ( 3/2 + < Hece 2a a a 2 = = 1 2a 2t + (pi 3/2 a t p ( i a (t ( 3/2 + 2 (pi 3 Case 2 a ( 3/2 (mod The, a 0 (mod Sice gcd(p, = 1, it follows that a b However, p β a b ad β τ < i This is a cotradictio Case 3 a ( 1/2 (mod It follows that 3a 3(pi 1 2 (pi (mod The fact that gcd(p, = 1 implies 3a b This cotradicts p γ 3a b ad γ < i Case 4 a t (mod with ( + 1/2 t < The 2a 2t (mod, 0 2t <, ad Hece a t ( 3/2 (mod, 0 t ( 3/2 < 2a a a 2 = 2a (2t pi + (pi 3/2 p ( i a (t ( 3/2 = 1 a t 2 (pi 3 Therefore, ν p (T (a, b, α τ, ad this implies ν p (3(a b(3a bt (a, b, α The proof of (2, for p 5, is complete Now assume p = 3 If 9, the ad It follows that α = 1, ad the (2 clearly holds If 9, the the proof of the case p 5 applies to this situatio I Case 2, a 0 (mod 3 i gives 3 i1 ab I Case 3, 3 i1 3ab Thus, if i τ +2, the i 1 τ + 1 It is a cotradictio i both cases Hece 2a a a 2 = 1 3 i 3 i 3 i 3 i 3 i the electroic joural of combiatorics 22(1 (2015, #P19 4

5 for i = τ + 2, τ + 3,, α It follows that ν 3 (T (a, b, α τ 1, ad the ν 3 (3(a b(3a bt (a, b, α That is, (2 also holds Hece, (a b(3a bt (a, b, Therefore, ( ( 2 2a (2 + 1( (a b(3a b a This completes the proof of Theorem 1 ( a Ackowledgemets We are grateful to the aoymous referee for carefully readig our origial mauscript ad givig may detailed commets Refereces [1] J W Bober Factorial ratios, hypergeometric series, ad a family of step fuctios J Lod Math Soc, 79: , 2009 [2] N J Calki Factors of sums of powers of biomial coefficiets Acta Arith, 86:17 26, 1998 [3] H-Q Cao, H Pa Factors of alteratig biomial sums J Adv i Appl Math, 45:96 107, 2010 [4] N J Fie Biomial coefficiets modulo a prime Amer Math Mothly, 54: , 1947 [5] V J W Guo Proof of Su s cojecture o the divisibility of certai biomial sums Electro J Combi, 20(4:#P420, 2013 [6] V J W Guo Proof of two divisibility properties of biomial coefficiets cojectured by Z-W Su Electro J Combi, 21(2:#P254, 2014 [7] V J W Guo, F Jouhet ad J Zeg, Factors of alteratig sums of products of biomial ad q-biomial coefficiets Acta Arith, 127:17 31, 2007 [8] V J W Guo, C Krattethaler Some divisibility properties of biomial ad q- biomial coefficiets J Number Theory, 135: , 2014 [9] V J W Guo, J Zeg Factors of biomial sums from the Catala triagle J Number Theory, 130: , 2010 [10] V J W Guo, J Zeg Factors of sums ad alteratig sums ivolvig biomial coefficiets ad powers of itegers It J Number Theory, 7: , 2011 [11] M Razpet O divisibility of biomial coefficiets Discrete Math, 135: , 1994 the electroic joural of combiatorics 22(1 (2015, #P19 5

6 [12] Z-W Su O divisibility of biomial coefficiets J Austral Math Soc, 93: , 2012 [13] Z-W Su Products ad sums divisible by cetral biomial coefficiets Electro J Combi, 20(1:#P19, 2013 the electroic joural of combiatorics 22(1 (2015, #P19 6