Further Results on a Curious Arithmetic Function
Let be an odd prime number and be a positive integer. Let , , and denote the -adic valuation of the integer , the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function defined by for any positive integer . We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the -adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.
Let be any given prime and be a positive integer. As usual, let , , and denote the -adic valuation of the integer , the set of positive integers, and the set of positive rational numbers, respectively. Arithmetic functions are classical topics in number theory (see, for example, [1–3]). Recent years, some kinds of periodic functions were introduced by some authors for the investigation of the least common multiple of integer sequences (see, for instance, [4–9]). In this paper, we introduce an interesting arithmetic function defined byfor any positive integer . We have, for example, etc. It is clear that is not always an integer for any given prime .
Throughout this paper, let always denote an odd prime. In this paper, we present several interesting arithmetic properties about that function and then use them to establish some curious results involving the -adic valuation. This paper is organised as follows. First in Section 2, we show that the product is always an integer for any positive integer , and it is, in fact, a multiple of all prime number not equal to and not exceeding . Subsequently in Section 3, we first give a lower bound for defined in Section 2. Also, two properties of -adic valuation are provided. From these results, we can show that does not exist for the odd prime , which is very different from the case of since Farhi  proved that . Finally, Section 4 is devoted to derive an upper bound of if . One notes that was studied by Farhi , and was investigated by Wang et al. . Actually, many results they obtained would be extended in this paper to the case of all odd primes .
In order to investigate the function , two auxiliary arithmetic functions and are needed, defined, respectively, byfor any integer . Note that the function is well defined since the product in the denominator of the right-hand side of (3) is actually finite because for any sufficiently large .
Let be a rational number. By , denote the -adic valuation of . Define by
So, for any positive integer . As usual, let be the integer-part function, i.e., represents the largest integer no more than , and represent the binomial coefficient. Here, we give a well-known fact, which will be used frequently in this paper, as follows.
Fact 1. Each of following is true.(i) for any and .(ii)(a).(b).(c) hold for any positive integers and .(iii)(Legendre’s formula) Let be a prime and be a positive integer. Then,where is the -adic representation of .
2. The Integrity of the Product Function
Let and for any positive integer . Now, we give the first result as follows.
Theorem 1. Let be any given prime number and be a positive integer. Then, the value is an integer.
Proof. First, by the definition of , for any positive integer , we haveThen,It follows thatSo, for all positive integers , we haveNow, by applying the function into each of two sides of (9), we obtainTherefore, to prove Theorem 1, we only need to prove that is an integer for any positive integer . For this purpose, by the definition of , one notes that for any positive integers and . Hence,Additionally, . Thus, is a multiple of the multinomial coefficient which is an integer. So, must be an integer. This finishes the proof of Theorem 1.
Next, let us discuss the divisibility of by . In fact, we have the following result.
Theorem 2. For any positive integer , is a multiple of . In particular, is a multiple of all prime numbers not equal to and not exceeding .
Proof. By (10) and (11), we know that to prove Theorem 2, it is sufficient to show that is a multiple of , which is equivalent to prove that for all prime numbers not exceeding , we havewhere . On the one hand, by (i) and (iii) in Fact 1, one hasOn the other hand, as , for each , one then getsBesides, is an integer. It then infers thatTherefore by (13) and (15), we havewhich implies that (12) is true. So, the proof of Theorem 2 is complete.
3. Properties of and
Let be any odd prime. Now, we turn our attention to the lower bound for . In this section, we first obtain a lower bound for . Secondly, two properties of -adic valuations will be given. Finally, one then concludes that does not exist, which varies from the result of the case obtained by Farhi  that . First of all, we present one lemma as follows.
Lemma 1. Let be a prime number. Let be integers with and . Then, for any positive integer , one has
Proof. We prove (17) by induction on as follows. For , (17) is clearly true. For the given integer , suppose that (17) holds for any integer . Let us now show that (17) is true for the integer . On the one hand, by Fact 1, we haveOn the other hand, by the assumption, we haveSo, combining (18) with (19), it then follows from Fact 1 thatas desired. This completes the proof of Lemma 1.
Let be a given prime number and be any positive integer. Now, we present a lower bound for .
Theorem 3. Let be an odd prime number and be a positive integer. Then, we have
Proof. Let be positive integer. First, we claim thatfor all primes with . In fact, for any positive integer , one hasIt follows thatwhich infers thatBy (11), the desired result (22) follows. Now, let us continue to prove inequality (21). It is well known that for each positive integer and each prime . Hence, for proving inequality (21), by (22), we only need to provewhich will be done in what follows.
By using a computer, we have checked that (26) is true for . Now, we show that (26) holds for any integer by induction on . Let be the fixed positive integer. Assume that (26) is true for all integers less than . We need to show that (26) is true for the integer . We divide our proof into the following three cases.
Case 1. , where is an integer with . In this case, we haveThen, by the assumption that and the facts that , we havesince holds for any real number . So, (26) is true if .
Case 2. , where is an integer with . Then, by Lemma 1 and the assumption, we derive thatBut one claims thatIn fact, let and be the derivative function of . Then, one checks that for all real numbers . It follows that increases monotonically over the interval . Then, . So, (30) holds. Hence, from (29) and (30), we have thatThis means that (26) is true if .
Case 3. , where is an integer with . In this case, similar to Case 2, we haveThis tells us that (26) holds if .
Combining the above cases, we have that (26) is true for the integer . So, by mathematical induction, we have that (26) holds for any integer . This completes the proof of Theorem 3.
In the following, we present two curious properties of -adic valuations from .
Theorem 4. Let be distinct primes and be a positive integer. Then, we have
Theorem 5. Let be a prime number and be a positive integer. Let be the -adic expansion of . Then, we have
Proof. First, it is easy to see that (37) is true when . So, in what follows, we let . Then, (37) is equivalent toNow, we show that (38) holds. Taking the -adic valuation on both sides of identity (9), using Fact 1 and (11), one hasOne notes that , where is an integer with . Then,Then, (39) becomesSinceit then follows thatHowever, by the -adic expansion of : , one hasThen, (43) together with (44) gives us thatwhich infers thatIt then follows from (iii) of Fact 1 thatTherefore, the desired result follows immediately from (41) and (47). The proof of Theorem 5 is complete.
Now, at the end of this section, by using Theorems 3 and 5, one gets a result of limitation concerning as follows.
Corollary 1. Let be an odd prime number. We have thatdoes not exist.
Proof. Let be an odd prime and be any positive integer. First, by (9) and Theorems 3 and 5, one haswhere is the -adic representation of . Now, taking with being a positive integer, one derives thatIt then follows thatSo, does not exist.
4. An Upper Bound for
Let be an odd prime. In this section, we give an upper bound for as follows.
Theorem 6. Let be an odd prime number and be a positive integer. We have that
Proof. First, we show thatfor all positive integers and odd primes . This will be done by dividing into two cases.
Case 4. . By the definition of and (ii) of Fact 1, one hasSo, in order to prove (53) holds for any prime , we only need to confirm thatfor in the case of , which will be easily done by taking logarithm into both sides of above inequality.
Case 5. . Let us prove (53) by induction on . First, it is easy to check (53) holds for . Assume (53) is true for the integer less than . Write , where and are integers with and . It then follows from (11) and Lemma 1 thatHence, by utilizing inequalities in (ii) of Fact 1, one derives thatSo, (57) gives us from the assumption thatIt follows that (53) holds for all integers with . Therefore, (53) is proved. Now, by using (10) and (53), one arrives atas desired. So, it completes the proof of Theorem 6.
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Conflicts of Interest
The authors declare that they have no conflicts of interest.
Kaimin Cheng provided the main idea of this paper and gave the proofs of the main theorems. Long Chen and Tingting Wang checked some details for the paper and did some work in writing this paper.
Kaimin Cheng would like to thank Professor Shuhong Gao for his warm hospitality and help during the former’s visit to Clemson University in the year of 2020. Kaimin Cheng was supported partially by China Scholarship Council Foundation (201908510050) and the Research Initiation Fund for Young Teachers of China West Normal University (412679). Long Chen was supported partially by Doctoral Research Initiation Fund Project of Panzhihua University.
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