The p -Adic Valuations of Sums of Binomial Coefficients

In this paper, we prove three supercongruences on sums of binomial coeﬃcients conjectured by Z.-W. Sun. Let p be an odd prime and let h ∈ Z with 2 h − 1 ≡ 0 ( mod p ) . For a ∈ Z + and p a > 3, we show that 􏽐 p a + 1 ) . Also, for any n ∈ Z + , we have ] p 􏽐 n − 1 k � hn k 􏼠 􏼡 2 kk 􏼠 􏼡 (− ( h /2 )) k 􏼠 􏼡 ≥ ] p ( n ) , where ] p ( n ) denotes the p -adic order of n . any integer m − 4 ) / p ) 􏽐 n − 1 k � 0 n − 1 k 􏼠 􏼡 2 kk 􏼠 􏼡

Also, for any n ∈ Z + , we have ] p n− 1 k�0 hn − 1 k

Introduction
Let p be an odd prime. In 2006, Pan and Sun [1] proved the congruence via a curious combinatorial identity. For any positive integer a and prime p ≥ 5, later Sun and Tauraso [2] established the following general result: Guo [3] conjectured a q-analogue of (2), which was confirmed by Liu and Petrov [4] using a q-analogue of Sun-Zhao congruence on harmonic sums and a q-series identity. Guo and Zudilin [5] also gave the q-generalizations of (2).

(3)
If p is an odd prime not dividing B, then it is known that p|u p− (Δ/p) (see, e.g., [6]). For a nonzero integer n and a prime p, let ] p (n) denote the p-adic valuation (or p-adic order) of n, i.e., ] p (n) is the largest integer such that p ] p (n) |n, especially ] p (0) � +∞ and we define ] p (m/n) � ] p (m) − ] p (n) for rational number m/n. For more developments on p-adic valuation, we refer the reader to the papers [7][8][9][10].
In 2011, Sun [11] proved that for any nonzero integer m and odd prime p with p∤m, there holds where Δ � m(m − 4). As a common extension of (4), Sun [12] showed that 1 pn and furthermore 1 n Let p be an odd prime and let m be an integer with p∤m. One can easily get the following formula: since for any . It looks like the left-hand side of (7) has some connection with the right-hand side. Motivated by (4) and (7), Sun [13] determined the sum where h is a p-adic integer and m ∈ Z with p∤m. For example, if h ≡ 0(modp) and (((2h ≡ 1(modp)) or p a > 3), then It is interesting to consider whether there exists the supercongruence as (8) modulo the higher powers of p in the case 2h − 1 ≡ 0(modp) and p a > 3. Sun [13] managed to investigate the above case and made the following conjecture. e first aim of this paper is to prove the conjectured results.

Theorem 1.
Let p be an odd prime and let h ∈ Z with 2h − 1 ≡ 0(modp). If a ∈ Z + and p a > 3, then Also, for any n ∈ Z + , we have On the other hand, based on (5) and (7), Sun [12] conjectured the corresponding result with p∤m. e second aim of this paper is to show the following result. For any integer m ≡ 0(modp) and positive integer n, we have e remainder of the paper is organized as follows. In the next section, we give some lemmas. e proofs of eorems 1 and 2 will be given in Section 3.

Some Lemmas
In the following section, for an assertion A, we adopt the notation: We know that [m � n] coincides with the Kronecker symbol δ m,n . Lemma 1. Let n, k, α be positive integers and p be a prime. en, Proof. Note that is lemma is a well-known congruence due to Osburn et al., see, e.g., [[15], (19)]. e following curious result is due to Sun [16].
By the above, we have completed the proof of Lemma 6.

Proofs of Theorems 1 and 2
Proof of eorem 1. We first prove (10). Let ] p (n) � a. (10) is evidently trivial when a � 0. Next, we suppose that a ≥ 1. With the help of (13), we have By (20) and (21), for any odd prime p, we get Repeating the above process a − 1 times, we obtain that Let us turn to (9). We assume that p ≥ 5. In view of (13), we obtain

Journal of Mathematics
For any positive integer a, we have With the help of Lemma 6, for a ≥ 2, we get If a � 1, by (20), (29), and (42), then e congruence (9) holds with a � 1 and p ≥ 5. If a ≥ 2, combining (42) and (43), we have modulo p a+1 , Repeating this process s − 1 times, we have By (20) and (29), we get At last, we only need to think about the case s � a − 1. From (45)-(47), we have So, we obtain (9) with a ≥ 2 and p ≥ 5. When a ≥ 2 and p � 3, the proof of (9) is very similar to the case p ≥ 5, only requiring a few additional discussions. Without loss of generality, it suffices to study the following sum in (46). Note that  (51) Recall that Strauss et al. [19] proved that for any positive integer n,