Counting certain quadratic partitions of zero modulo a prime number

Abstract: Consider an odd prime number ≡ ( ) p 2 mod 3 . In this paper, the number of certain type of partitions of zero in /p is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in /p with all three parts chosen from the set of non-zero quadratic residues mod p. Such partitions are divided into two types. Those with exactly two of the three parts identical are classified as type I. The type II partitions are those with all three parts being distinct. The number of partitions of each type is given. The problem of counting such partitions is well related to that of counting the number of non-trivial solutions to the Diophantine equation + + = x y z 0 2 2 2 in the ring /p . Correspondingly, solutions to this equation are also classified as type I or type II. We give the number of solutions to the equation corresponding to each type.


Introduction
The study of Diophantine equations is one of the oldest branches of number theory and the most famous Fermat's last conjecture states that for any integer > n 2, no three positive integers can satisfy the Fermat's equation. For more than 350 years, many mathematicians have devoted their work on solving this fascinating conjecture and related Diophantine equations. It was until 1995, the conjecture became Fermat's last theorem, proved by Andrew Wiles [1].
Research has been done in the investigation of Diophantine equations "locally," in particular, over finite fields. For example, in [2], solving a Diophantine equation modulo every prime number was studied using commutative algebra techniques. Recently, many scholars have studied Diophantine equations modulo prime numbers and obtained a series of interesting results (see [3][4][5][6][7]).
Consider any odd prime number p. In this paper, we focus on the following three equations concerning solutions in the ring /p : It is known that if ≡ p 2 (mod 3), every element ∈ / a p is a perfect cube in /p . Precisely, if = + p m 3 2, where ∈ m , = ( ) + a a m 2 1 3 . Also, ↔ x x 3 is a one-to-one and onto correspondence on /p . Thus, in the case of ≡ p 2 ( ) ∀ ∈ / a p mod 3 , , a is a perfect square if and only if a is a perfect sixth power. Furthermore, the number of solutions to equation (1) is the same as that of equation (3). We focus on the solution triples ( ) x y z , , with ≠ xyz 0 in /p and call such a solution a non-trivial solution. A non-trivial solution triple ( ) x y z , , is called a quadratic solution if x y z , , are all non-zero perfect squares in /p (or non-zero quadratic residues mod p). Obviously, a quadratic solution to equation (2) gives a non-trivial solution to equation (3) and vice versa. Thus, the number of non-trivial solutions to equation (1) is the same as the number of quadratic solutions to equation (2). In [8][9][10], some character sums involving Dirichlet characters were used to determine the number of solutions to certain Diophantine equations modulo a prime number. In this paper, we apply formulas developed in [8] to determine the number of non-trivial solutions to equation (1) or that of quadratic solutions to equation (2), modulo any odd prime ≡ p 2 (mod 3). Another way to describe a non-trivial solution ( ) x y z , , to equation (1) is to view it as a three-part partition of 0 modulo p. The theory of partition is another interesting branch of number theory. The concept of partitions of positive integers was given by Leonard Euler in the 18th century. Since then many prominent mathematicians, including Gauss, Jacobi, Schur, McMahon, Andrews, Ramanujan, and Hardy, have made great contributions to the study of partitions (see [11][12][13]). Applications of partitions of positive integers or sets can be found in many other areas such as combinatorics, computer science, and genetics. A partition of a positive integer n is a combination (unordered, with repetitions allowed) of positive integers, called the parts, that add up to n. A three-part partition of 0 in /p has the form of = + + x y z 0 in /p , where x y z , , are all non-zero elements in /p . We are interested in counting the number of three-part partitions of 0 in /p , where each part is a non-zero quadratic residue modulo p. We call such a partition of 0 a 3Q-partition of 0. For any 3Q-partition of 0 with three distinct parts, it produces six non-trivial solutions to equation (1) (six ordered solution triples). Likewise, if two of the three parts in a 3Q-partition of 0 are identical, then it gives three non-trivial solutions to equation (1). Thus, the number of 3Q-partitions of 0 is well related to the number of non-trivial solutions to equation (1), which is the same as the number of non-trivial quadratic solutions to equation (2), and can be determined consequently. We discuss these two different types of 3Q-partitions of 0 and the enumeration for each type. We start with the following definitions. , , and 0 in .
, , to equation (2) or to the equation Note that a 3Q-partition + + = a b c 0 in /p is represented by the set { } a b c , , which may be a multiset because two of the three numbers may be identical. Next, we adopt a definition originated in [8].
for every odd prime p. As mentioned previously, if ≡ ( ) p 2 mod 3 , there is a one-to-one and onto correspondence between S p and the set of quadratic solutions to equation (2). Thus, the size of S p is the number of quadratic solutions to equation (2), which is also the number of quadratic solutions to + + = x y z 0 in /p . It implies that | | = ( In Next, we focus on counting the number of related restricted partitions of 0. Assume { } a b c , , is a 3Qpartition of 0 in /p . That is, ∈ a b c r , , p and + + = ( ) a b c p 0 mod . Because p is a prime greater than 3, only two cases may occur: either exactly two out of the three numbers are identical, or all three are distinct. We divide the 3Q-partitions of 0 into two types based on this distinction. Recall that ( ) X p denotes the set of all 3Q-partitions of 0 in /p represented by multisets.  , that is, ( In Section 3, we analyze quadratic solution types to equation (2) and give the exact formulas, respectively, for α p and β p . The main result is given in Theorem 3.8. In Section 4, we use the solution triples in S p to build monomials and homogeneous polynomials in the polynomial ring ( / )[ ] p x y z , , . We give a formula for the number of such polynomials.

Number of solutions
In this section, we calculate the number of non-trivial solutions to equation (1) or equation (3), or the number of quadratic solutions to equation (2), by applying properties of Dirichlet characters (see [14][15][16][17][18]) and the character sum    That is, Proof. By Lemma 2.1, there is a one-to-one and onto correspondence between the quadratic solutions ( ) ∈ a b c r , , to the equation In [8], an explicit formula for ( ) A p k is given: Let p be an odd prime with ≡ ( ) p 2 mod 3 . Then for every positive integer k, the Dirichlet character sum ( ) A p k is given by . We use this value to help evaluating ( ) Proof. Since ≡ ( ) p 2 mod 3 , when a passes through a reduced residue system mod p, a 3 also passes through a reduced residue system mod p. Note that the equation + + = a b c 0 and involved in the proof below is over the ring /p .
We expand the above expression into a sum of eight parts and further into a sum with four summands: The last summand is 0 by Lemma 2. . That is, there are 24 triples from r 17 3 whose elements add up to 0 modulo 17. They are given as follows:  , 4 , 4, 4, 9 , 9, 4, 4 , 9, 16, 9 , 16, 9, 9 , 9, 9, 16   in /p , is similar. It can be evaluated as follows.
The second equality above involves steps similar to those in the proof of Theorem 2.4. We skip the details. □ Research has been done on the distribution and density of solutions to a Diophantine equation in the set of all the possible elements. For example, in [19], the number ( ) N p of the integer solutions modulo a prime p to the equation = − y x x 2 5 is studied. It is shown that | ( ) − | N p p is relatively small compared to p. More precisely, | ( ) − | < N p p p 4 and the constant 4 is the best possible. In our study, all the nontrivial solutions to equation (1) are chosen from the set r p 3 whose size is ( − ) / p 1 8 3 . Table 1 shows the ratio of the number of solutions to the size of r p 3 for the first six primes which reflects the density of the solutions in the set r p 3 .
It seems when p is larger and larger, the density is smaller and smaller. We define the density function as follows.  In the next theorem, we give the explicit formula for the density function which indicates that when p is approaching to infinity, the density function acts similarly as /p 1 . Note that, when ≡ ( ) p 2 mod 3 , ≡ p 5, 11, 17, or 23 modulo 24. It confirms that the density function ( ) D p is approaching to 0 when p goes to infinity. Actually, ( ) = ( / ) D p p 1 .

Solution types and the resulting partitions
A classical number theory problem is on the partitions of positive integers. Many researchers have studied counting the number of integers partitions and that of restricted partitions, conjugate and self-conjugate partitions, graphical representations of partitions, and so on. There are many applications of integer partitions in other fields such as molecular chemistry, crystallography, and quantum mechanics. In modern algebra, it is well related to the study of symmetric polynomials, which we will see in Section 4.
Let p be a prime number with ≡ ( ) p 2 mod 3 .  . For any prime ≡ ( ) p 2 mod 3 which is greater than 5, | | ≠ S 0 p . Next we consider primes p greater than 5 with ≡ ( ) p 2 mod 3 . For such a prime number p, ≠ ∅ S p and in /p , ≠ a 3 0 if ≠ a 0. We examine the quadratic triples ( ) ∈ a b c S , , p . Among these triples, only two cases may occur: (1) two of the three are identical but not all the same or (2) all three numbers a b c , , are distinct. By Definition 1.5, these two cases correspond to solutions of type I or type II. We concern the number of triples in S p of each type. Later we show that, for certain prime numbers p, S p has no triples of type II.
We note that in Table 2, | | = | | S S 1 2 for some primes such as 17, 29, 41, 53, and so on. We classify the prime numbers achieving this property as follows.