On Pythagorean Triples and the Primitive Roots Modulo a Prime

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 p be an odd prime large enough. Then, there must exist three primitive roots x, y , and z modulo p such that x 2 + y 2 � z 2


Introduction
It is well known that all positive integer solutions of the equation x 2 + y 2 � z 2 are x � t(a 2 − b 2 ), y � 2tab, and z � t(a 2 + b 2 ), where t, a, and b are arbitrary positive integers satisfying a > b; a and b have no prime divisors in common, and one of a or b is odd and the other is even. is result can be found in many elementary number theory textbooks, e.g., reference [1] (see eorems [2][3][4][5][6][7][8][9]. We call such a set of solution (x, y, z) as a Pythagorean triple. For example, taking t � b � 1 and a � 2 or 4, then (x, y, x) � (3,4,5), (15,8,17) are two Pythagorean triples.
In this paper, we only focus on the solutions of the form where a and b are any positive integers. On the other hand, let p be a fixed odd prime and g be an integer with (g, p) � 1. If g, g 2 , . . . , g p− 1 form a reduced residue system modulo p, then g is called a primitive root modulo p. For other properties of the primitive roots and related results, see references [1][2][3][4][5][6][7][8][9][10][11], which we will not cover here. In this paper, we will consider the following problem.
For any odd prime p and integer 1 < M < p, whether there is a Pythagorean triple (x, y, z) in (1) with 1 ≤ a, b ≤ M, such that x, y, and z all are the primitive roots modulo p?
If there are, let C(M, p) denotes the number of all such Pythagorean triples (x, y, z) in (1) with 1 ≤ a, b ≤ M. en, how does C(M, p) depend on p?
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 p, D. H. Lehmer numbers, and Lucas and Fibonacci sequences.
In this paper, let us make two simple conclusions about the asymptotic properties of C(M, p) as follows.

Theorem 1. For any odd prime p, we have the asymptotic formula
where, as usual, ϕ(n) denotes the Euler function and ω(n) denotes the number of all distinct prime divisors of n.

Theorem 2.
Let p be an odd prime with p ≡ 1 mod 4. en, for any integer 1 < M ≤ p, we have the asymptotic formula It is clear that for any positive number ε > 0, if M > p 1/2+ε , then eorem 2 is nontrivial. at is, the main term is larger than the error terms.
From our theorems, we may immediately deduce the following two corollaries. Corollary 1. Let p be an odd prime large enough. en, there must exist three primitive roots x, y, and z modulo p such that Corollary 2. Let p ≡ 1 mod 4 be a prime large enough. en, for any integer p 1/2+ε < M < p, there must exist three primitive roots x, y, and z modulo p with 1 ≤ x, y, z ≤ M 2 such that where ε > 0 is any fixed positive number.

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 p be an odd prime. en, for any integer a with (a, p) � 1, we have the identity where e(y) � e 2πiy , k r�1 ′ denotes the summation over all integers 1 ≤ r ≤ k such that r is coprime to k, μ(n) is the Möbius function, and ind(a) denotes the index of a relative to some fixed primitive root g mod p. Proof Proof.
is is Lemma 17 in [14]. Some related work can also be found in [15]. □ Lemma 3. Let p be an odd prime. en, for any characters χ 1 , χ 2 , and χ 3 (not all the principal characters) modulo p, we have the estimate Proof. If χ 2 is an odd character modulo p, then which implies If χ 2 is an even character modulo p, then there is a character ψ modulo p such that χ 2 � ψ 2 . Now, from the properties of the Legendre symbol modulo p and Lemma 2, we have the estimate If χ 1 and χ 3 are the principal character modulo p, then χ 2 is not the principal character modulo p. In this case, we have Combining (10) Proof. For any integer n, from the trigonometric identity From the properties of the reduced residue system, complete residue system modulo p, and Lemma 3, we have Journal of Mathematics Note that if p ≡ 1mod4, then there exist two integers u 0 and − u 0 such that the congruence 4x 2 ≡ − 1modp. So we have the method of proving (16), and the properties of Gauss sums, we have where τ(χ) � p− 1 a�1 χ(a)e(a/p) denotes the classical Gauss sums and |τ(χ)| ≤ � � p √ .
Similarly, we also have and where a denotes the solution of the congruence equation ax ≡ 1modp.

Proofs of the Theorems
In this section, we shall complete the proofs of our main results. First, we prove eorem 1. From the definition of C(p, p), Lemma 1, and the properties of the complete residue system modulo p, we have the identity