Abstract
In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let be an odd prime large enough. Then, there must exist three primitive roots , and modulo such that .
1. Introduction
It is well known that all positive integer solutions of the equation are , , and , where , , and are arbitrary positive integers satisfying ; and have no prime divisors in common, and one of or is odd and the other is even. This result can be found in many elementary number theory textbooks, e.g., reference [1] (see Theorems 2–9). We call such a set of solution as a Pythagorean triple. For example, taking and or , then are two Pythagorean triples.
In this paper, we only focus on the solutions of the formwhere and are any positive integers.
On the other hand, let be a fixed odd prime and be an integer with . If form a reduced residue system modulo , then is called a primitive root modulo . For other properties of the primitive roots and related results, see references [1–11], which we will not cover here. In this paper, we will consider the following problem.
For any odd prime and integer , whether there is a Pythagorean triple in (1) with , such that , and all are the primitive roots modulo ?
If there are, let denotes the number of all such Pythagorean triples in (1) with . Then, how does depend on ?
We think these problems are interesting, and they depict the distribution properties of the Pythagorean triples in other special integer sets, such as the primitive roots modulo , D. H. Lehmer numbers, and Lucas and Fibonacci sequences.
In this paper, let us make two simple conclusions about the asymptotic properties of as follows.
Theorem 1. For any odd prime , we have the asymptotic formulawhere, as usual, denotes the Euler function and denotes the number of all distinct prime divisors of .
Theorem 2. Let be an odd prime with . Then, for any integer , we have the asymptotic formula
It is clear that for any positive number , if , then Theorem 2 is nontrivial. That is, the main term is larger than the error terms.
From our theorems, we may immediately deduce the following two corollaries.
Corollary 1. Let be an odd prime large enough. Then, there must exist three primitive roots , and modulo such that
Corollary 2. Let be a prime large enough. Then, for any integer , there must exist three primitive roots , and modulo with such thatwhere is any fixed positive number.
2. Several Lemmas
To complete the proof of our main result, we need the following four simple lemmas. For the sake of simplicity, we do not repeat some elementary number theory and analytic number theory results, which can be found in references [10, 12, 13]. First, we have the following.
Lemma 1. Let be an odd prime. Then, for any integer with , we have the identitywhere , denotes the summation over all integers such that is coprime to , is the Möbius function, and denotes the index of relative to some fixed primitive root .
Proof. See Proposition 2.2 in [13].
Lemma 2. Let be an odd prime and be Dirichlet characters modulo , at least one of which is nonprincipal character. Let be an integral coefficient polynomial of degree . Then, for pairwise distinct integers , we have the estimate
Proof. This is Lemma 17 in [14]. Some related work can also be found in [15].
Lemma 3. Let be an odd prime. Then, for any characters , , and (not all the principal characters) modulo , we have the estimate
Proof. If is an odd character modulo , then , so we havewhich impliesIf is an even character modulo , then there is a character modulo such that . Now, from the properties of the Legendre symbol modulo and Lemma 2, we have the estimateIf and are the principal character modulo , then is not the principal character modulo . In this case, we haveCombining (10)–(12), we may immediately deduce Lemma 3.
Lemma 4. Let be an odd prime with and be an integer. Then, for any characters , , and (not all the principal characters) modulo , we have the estimate
Proof. For any integer , from the trigonometric identitywe haveFrom the properties of the reduced residue system, complete residue system modulo , and Lemma 3, we haveNote that if , then there exist two integers and such that the congruence . So we haveFrom the estimatethe method of proving (16), and the properties of Gauss sums, we havewhere denotes the classical Gauss sums and .
Similarly, we also haveandwhere denotes the solution of the congruence equation .
Combining (15), (16), and (19)–(21), we have the estimateThis proves Lemma 4.
3. Proofs of the Theorems
In this section, we shall complete the proofs of our main results. First, we prove Theorem 1. From the definition of , Lemma 1, and the properties of the complete residue system modulo , we have the identitywhere denotes a Dirichlet character modulo .
If is the principal character modulo , then we have
If one of , , or is a nonprincipal character modulo , then from Lemma 3 and the properties of the reduced residue system modulo , we have
Note that the estimatewhere denotes the number of all distinct prime divisors of .
Combining (23), (27), and (28), we have the asymptotic formula
This proves Theorem 1.
Similarly, from Lemma 4 and the method of proving Theorem 1, we can also deduce Theorem 2. Details are not given in this study.
4. Conclusion
The main results of this paper are two theorems, which are closely related to Pythagorean triples and primitive roots modulo an odd prime . It describes that when the prime is large enough, then there must exist three primitive roots , and modulo such that . At the same time, we also give a sharp asymptotic formula for the counting function of all such solutions . Of course, our conclusion can also be generalized to other special integer sets, such as D. H. Lehmer numbers and -th residue modulo .
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
The author contributed to the work and read and approved the final manuscript.
Acknowledgments
This work was supported by the NSF of China (11862018) and NSF of Inner Mongolia of China (2017BS0101).