Abstract
In this paper, we prove three supercongruences on sums of binomial coefficients conjectured by Z.-W. Sun. Let be an odd prime and let with . For and , we show that . Also, for any , we have , where denotes the -adic order of . For any integer and positive integer , we have , where is the Legendre symbol and is the ring of -adic integers.
1. Introduction
Let be an odd prime. In 2006, Pan and Sun [1] proved the congruencevia a curious combinatorial identity. For any positive integer and prime , later Sun and Tauraso [2] established the following general result:
Guo [3] conjectured a -analogue of (2), which was confirmed by Liu and Petrov [4] using a -analogue of Sun–Zhao congruence on harmonic sums and a -series identity. Guo and Zudilin [5] also gave the -generalizations of (2).
Let . The Lucas sequence is given by
If is an odd prime not dividing , then it is known that (see, e.g., [6]). For a nonzero integer and a prime , let denote the -adic valuation (or -adic order) of , i.e., is the largest integer such that , especially and we define for rational number . For more developments on -adic valuation, we refer the reader to the papers [7–10].
In 2011, Sun [11] proved that for any nonzero integer and odd prime with , there holdswhere . As a common extension of (4), Sun [12] showed thatand furthermore
Let be an odd prime and let be an integer with . One can easily get the following formula:since for any , we get . 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 modulo , where is a -adic integer and with . For example, if and ((() or ), then
It is interesting to consider whether there exists the supercongruence as (8) modulo the higher powers of in the case and . Sun [13] managed to investigate the above case and made the following conjecture. The first aim of this paper is to prove the conjectured results.
Theorem 1. Let be an odd prime and let with . If and , then
Also, for any , we have
On the other hand, based on (5) and (7), Sun [12] conjectured the corresponding result with . The second aim of this paper is to show the following result.
Theorem 2. Let be an odd prime and let . For any integer and positive integer , we have
The remainder of the paper is organized as follows. In the next section, we give some lemmas. The proofs of Theorems 1 and 2 will be given in Section 3.
2. Some Lemmas
In the following section, for an assertion , we adopt the notation:
We know that coincides with the Kronecker symbol .
Lemma 1. Let be positive integers and be a prime. Then,
Proof. Note thatThis proves (13). The congruence (13) is a result of Beukers [[14], Lemma 2].
Lemma 2. Let be an odd prime. Then, for any integers and positive integers , we have
This lemma is a well-known congruence due to Osburn et al., see, e.g., [[15], (19)].
The following curious result is due to Sun [16].
Lemma 3 (see [16], Theorem 1]). Let and . Suppose that is an odd prime dividing . Then,
Furthermore,and alsowhere denotes the Catalan number . Thus, for , we have
Lemma 4. Let be an odd prime and let with . Let be nonnegative integers. If , then we have
If , then
If and , then
Proof. Observe thatSince and , by (17), we obtain thatIf , by (15) and (24), we getThus, (20) is proved. The congruence (21) is easily deduced from (24).
Finally, we will prove (22). The congruence (22) is trivial when . Now we may assume . With the help of (15), (17), and (24), for any nonnegative integer with , we haveThis concludes the proof.
Lemma 5. Let be an odd prime and be nonnegative integers. Let . For any integer , we have
Proof. The proof is very similar to (20). Clearly,Substituting and in (6), we obtain (27).
Lemma 6. Let be an odd prime and with . Let be a positive integer. Then,
Proof. Note thatWith the help of Lucas’ theorem (cf. [17], p. 44]), it follows thatSince , in view of (16), thenThe congruence (29) with holds by (31) and (32).
Now suppose that . In fact, for , we clearly haveObserving that , we obtainRecall the Chu–Vandermonde identity (see, e.g., [18], p. 169])Thus,Therefore, (29) with is proved by (31), (34), and (36).
By the above, we have completed the proof of Lemma 6.
3. Proofs of Theorems 1 and 2
Proof of Theorem 1. We first prove (10). Let . (10) is evidently trivial when . Next, we suppose that . With the help of (13), we haveBy (20) and (21), for any odd prime , we getRepeating the above process times, we obtain thatLet us turn to (9). We assume that . In view of (13), we obtainFor any positive integer , we haveTherefore,With the help of Lemma 6, for , we getIf , by (20), (29), and (42), thenThe congruence (9) holds with and . If , combining (42) and (43), we have modulo ,Repeating this process times, we haveBy (20) and (29), we getAt last, we only need to think about the case . From (45)–(47), we haveSo, we obtain (9) with and . When and , the proof of (9) is very similar to the case , only requiring a few additional discussions. Without loss of generality, it suffices to study the following sum in (46). Note thatWith the help of (22), we obtainIn light of Lucas’ theorem,Recall that Strauss et al. [19] proved that for any positive integer ,Guo and Zeng [20] deduced a partial -analogue of (52). Combining (50)–(52), we have modulo ,Since , then . From (45), (46), and (53),Proof of Theorem 1 is now complete.
Proof of Theorem 2. Let with and . The proof of (11) is very similar to (10). With the help of (13), for any integer and nonnegative integer , now we haveTherefore,In light of Lemma 5, we haveBy Lemma 5 and (55), we getwhere the last result comes from Lemma 5.
In view of the above, we have completed the Proof of Theorem 2.
Data Availability
No data were used to support this study.
Disclosure
An earlier version of this article has been presented as preprint on arXiv which can be accessed from the following link: https://arxiv.org/abs/1911.00005.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (grant nos. 11971222 and 12071208), Natural Science Foundation of Nanjing Institute of Technology (no. CKJB201807), and Special Program for Student Innovation and Entrepreneurship Training (S20190025).