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Counting certain quadratic partitions of zero modulo a prime number

  • Wang Xiao EMAIL logo and Aihua Li
Published/Copyright: May 3, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

Consider an odd prime number p ≡ 2 ( mod 3 ) . In this paper, the number of certain type of partitions of zero in Z / p Z is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in Z / p Z 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 2 + y 2 + z 2 = 0 in the ring Z / p Z . 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.

Keywords: partition of a number; Dirichlet character sum; Diophantine equation; solution number
MSC 2010: 11D45; 11P83; 11L10

1 Introduction

The study of Diophantine equations is one of the oldest branches of number theory and the most famous Diophantine equation is the Fermat equation: x n + y n = z n . Conjectured by Pierre de Fermat in 1637, 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 Z / p Z :

( 1 ) x 2 + y 2 + z 2 = 0 , ( 2 ) x 3 + y 3 + z 3 = 0 , ( 3 ) x 6 + y 6 + z 6 = 0 .

It is known that if p ≡ 2 (mod 3), every element a ∈ Z / p Z is a perfect cube in Z / p Z . Precisely, if p = 3 m + 2 , where m ∈ Z , a = ( a 2 m + 1 ) 3 . Also, x ↔ x 3 is a one-to-one and onto correspondence on Z / p Z . Thus, in the case of p ≡ 2   ( mod 3 ) , ∀ a ∈ Z / p Z , 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 x y z ≠ 0 in Z / p Z 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 Z / p Z (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 Z / p Z has the form of 0 = x + y + z in Z / p Z , where x , y , z are all non-zero elements in Z / p Z . We are interested in counting the number of three-part partitions of 0 in Z / p Z , 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.

Definition 1.1

Let p be any odd prime number.

  1. Denote r p = { the non-zero quadratic residues modulo p } .

  2. S p = { ( a , b , c ) ∣ a , b , c ∈ r p and a + b + c = 0 in Z / p Z } .

  3. A triple ( a , b , c ) ∈ ( Z / p Z ) 3 is called non-trivial if a b c ≠ 0 in Z / p Z . A solution triple ( a , b , c ) to equation (2) or to the equation x + y + z = 0 in Z / p Z is called a quadratic solution if a , b , c ∈ r p .

  4. A multiset { a , b , c } is called a 3Q-partition of 0 in Z / p Z if a , b , c ∈ r p and a + b + c = 0 . Denote X ( p ) = { all 3Q-partitions of 0 in Z / p Z } .

Note that a 3Q-partition a + b + c = 0 in Z / p Z 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].

Definition 1.2

[8] Given positive integers h and k and a prime number p > 2 , N ( h , k , p ) denotes the number of non-trivial solutions to the equation

x 1 h + ⋯ + x k h = 0 in Z / p Z , where x 1 , … , x k ∈ r p .

Remark 1.3

It is known that ∣ r p ∣ = ( p − 1 ) / 2 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 Z / p Z . It implies that ∣ S p ∣ = N ( 3 , 3 , p ) = N ( 1 , 3 , p ) .

In [8], a character sum A k ( p ) , where k is a positive integer, was studied and it is useful in calculating the size of S p in this study. Originally, it was defined for any Dirichlet character χ mod p . We adopt this definition with respect to the special Dirichlet character ∗ p , which is the Legendre symbol mod p .

Definition 1.4

Let p be an odd prime and k be a positive integer. The Dirichlet character sum A k ( p ) is defined by

A k ( p ) = ∑ a 1 = 1 p − 1 ∑ a 2 = 1 p − 1 ⋯ ∑ a k = 1 p − 1 a 1 3 + a 2 3 + ⋯ + a k 3 = 0 in Z / p Z a 1 p a 2 p ⋯ a k p .

We consider the case when p ≡ 2 ( mod 3 ) . The above sum function A k ( p ) plays an important role in determining the number of solutions to equation (1) or the number of 3Q-partitions of 0 in Z / p Z . In Section 2, we apply the results about A k ( p ) in [8] to develop the exact number N ( 3 , 3 , p ) = N ( 1 , 3 , p ) (Theorem 2.4). Similarly, N ( 3 , 2 , p ) and N ( 3 , 4 , p ) are calculated. At the end of Section 2, we discuss the density of solution triples over the set r p 3 .

Next, we focus on counting the number of related restricted partitions of 0. Assume { a , b , c } is a 3Q-partition of 0 in Z / p Z . That is, a , b , c ∈ r p and a + b + c = 0 ( mod p ) . 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 Z / p Z represented by multisets.

Definition 1.5

Let p be an odd prime.

  1. X 1 ( p ) = { { x , x , y } ∣ x , y ∈ r p , x ≠ y , and x + x + y = 0 in Z / p Z } . Any element in X 1 ( p ) is called a 3Q-partition of 0 in Z / p Z of type I.

  2. X 2 ( p ) = { 3Q-partitions { x , y , z } of 0 in Z / p Z , where x , y , z are all distinct } . Elements in X 2 ( p ) are called 3Q-partitions of 0 in Z / p Z of type II.

  3. Correspondingly, a solution triple ( a , b , c ) in Z / p Z to any equation is of type I if exactly two of the three numbers are identical. It is of type II if all the three numbers are distinct.

  4. Denote α p = ∣ X 1 ( p ) ∣ and β p = ∣ X 2 ( p ) ∣ .

In other words, α p is the number of 3Q-partitions of 0 of type I and β p is the number of 3Q-partitions of 0 of type II.

Remark 1.6

Obviously, α p + β p = ∣ X ( p ) ∣ . Assume x + x + y = 0 is a 3Q-partition of 0 of type I in Z / p Z . It produces three elements in S p : ( x , x , y ) , ( x , y , x ) , and ( y , x , x ) . Similarly, a 3Q-partition of type II produces six triples in S p by permutation. Thus, ∣ S p ∣ = 3 ∣ X 1 ( p ) ∣ + 6 ∣ X 2 ( p ) ∣ , that is, N ( 3 , 3 , p ) = N ( 1 , 3 , p ) = 3 α p + 6 β p .

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 ( Z / p Z ) [ x , y , z ] . We give a formula for the number of such polynomials.

2 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 A k ( p ) defined in Definition 1.4. We focus on prime numbers p in the form of p = 3 m + 2 , where m is a positive integer. Combinatorial and number theory methods are used to derive an explicit formula for the solution number ∣ S p ∣ = N ( 3 , 3 , p ) = N ( 1 , 3 , p ) .

The following lemma shows a known result in the literature. For convenience, we re-state it and provide a simple proof.

Lemma 2.1

Let p be a prime number in the form of p = 3 m + 2 , where m is a positive integer. Then every element in Z / p Z is a cubic residue mod p .

Proof

Let a ∈ Z / p Z . By Fermat’s little theorem a p ≡ a ( mod p ) . Thus, a ≡ a p ⋅ a p − 1 = a 2 p − 1 = ( a 2 m + 1 ) 3 ( mod p ) . Therefore, every element in Z / p Z is a cubic residue mod p .□

Consequently, we claim

Lemma 2.2

Let p be a prime number in the form of p = 3 m + 2 , where m is a positive integer. Then

∣ S p ∣ = ∣ { ( a , b , c ) ∣ a , b , c ∈ r p and a 3 + b 3 + c 3 = 0 in Z / p Z } ∣ = ∣ { ( a , b , c ) ∣ a , b , c ∈ Z / p Z , a b c ≠ 0 , and a 6 + b 6 + c 6 = 0 in Z / p Z } ∣ = ∣ { ( a , b , c ) ∣ a , b , c ∈ r p and a + b + c = 0 in Z / p Z } ∣ = ∣ { ( a , b , c ) ∣ a , b , c ∈ Z / p Z , a b c ≠ 0 , and a 2 + b 2 + c 2 = 0 in Z / p Z } ∣ .

That is, ∣ S p ∣ = N ( 3 , 3 , p ) = N ( 1 , 3 , p ) .

Proof

By Lemma 2.1, there is a one-to-one and onto correspondence between the quadratic solutions ( a , b , c ) ∈ r p 3 to the equation x + y + z = 0 in Z / p Z and the quadratic solutions ( a 2 m + 1 , b 2 m + 1 , c 2 m + 1 ) ∈ r p 3 to the equation x 3 + y 3 + z 3 = 0 in Z / p Z .□

In [8], an explicit formula for A k ( p ) is given:

Lemma 2.3

[8] Let p be an odd prime with p ≡ 2 ( mod 3 ) . Then for every positive integer k , the Dirichlet character sum A k ( p ) is given by

A k ( p ) = 0 if k is odd ; ( − 1 ) k ( p − 1 ) / 4 ⋅ ( p − 1 ) ⋅ p ( k − 2 ) / 2 if k is even .

In particular, A 3 ( p ) = 0 . We use this value to help evaluating N ( 3 , 3 , p ) or N ( 1 , 3 , p ) .

Theorem 2.4

If p is an odd prime with p ≡ 2 ( mod 3 ) , then

∣ S p ∣ = N ( 3 , 3 , p ) = N ( 1 , 3 , p ) = 1 8 ( p − 1 ) p − 2 + 3 ( − 1 ) p + 1 2 .

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 a 3 + b 3 + c 3 = 0 involved in the proof below is over the ring Z / p Z .

N ( 3 , 3 , p ) = 1 8 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a 3 + b 3 + c 3 = 0 1 + a p 1 + b p 1 + c p = 1 8 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 1 + a p 1 + b p 1 + c p = N ( 1 , 3 , p ) .

We expand the above expression into a sum of eight parts and further into a sum with four summands:

N ( 1 , 3 , p ) = 1 8 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 1 + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 b p + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 c p + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p b p + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p c p + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 b p c p + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p b p c p = 1 8 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 1 + 3 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p + 3 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p b p + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p b p c p .

Furthermore, if a + b + c = 0 with a b c ≠ 0 in Z / p Z , then b ≠ − a and c = − ( a + b ) in Z / p Z . Thus, in the above sum, the first of the four summands is calculated as follows:

∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 1 = ( p − 1 ) ( p − 2 ) .

Note that when a passes through a reduced residue system mod p , a b also passes through a reduced residue system mod p for all b with 1 ≤ b ≤ p − 1 . Then we replace a by a b and c by c b and apply properties of Dirichlet character sums to calculate the second and the third summand:

∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + 1 + c = 0 a p b p = 0

and

∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p b p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + 1 + c = 0 a p = ( p − 1 ) ∑ a = 1 p − 1 ∑ c = 1 p − 1 a + 1 + c = 0 a p = ( p − 1 ) 0 − − 1 p = ( p − 1 ) ( − 1 ) p + 1 2 .

The last summand is 0 by Lemma 2.3:

A 3 ( p ) = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 a + b + c = 0 a p b p c p = 0 .

Finally, from all the above,

N ( 1 , 3 , p ) = 1 8 ( p − 1 ) ( p − 2 ) + 3 ( p − 1 ) ( − 1 ) p + 1 2 = p − 1 8 p − 2 + 3 ( − 1 ) p + 1 2 = ( p − 1 ) ( p − 5 ) 8 , if p ≡ 1 ( mod 4 ) , ( p − 1 ) ( p + 1 ) 8 , if p ≡ 3 ( mod 4 ) . □

Example 2.5

Consider p = 17 . The set of non-zero quadratic residues mod 17 has eight numbers:

r 17 = { 1 , 2 = 6 2 , 4 , 8 = 5 2 , 9 , 13 = 8 2 , 15 = 7 2 , 16 } .

By Theorem 2.4, ∣ S 17 ∣ = N ( 1 , 3 , 17 ) = ( 1 / 8 ) ( 17 − 1 ) ( 17 − 2 + 3 ( − 1 ) 9 ) = 24 . That is, there are 24 triples from r 17 3 whose elements add up to 0 modulo 17. They are given as follows:

S 17 = { ( 1 , 1 , 15 ) , ( 1 , 15 , 1 ) , ( 15 , 1 , 1 ) , ( 1 , 8 , 8 ) , ( 8 , 8 , 1 ) , ( 8 , 1 , 8 ) , ( 4 , 9 , 4 ) , ( 4 , 4 , 9 ) , ( 9 , 4 , 4 ) , ( 9 , 16 , 9 ) , ( 16 , 9 , 9 ) , ( 9 , 9 , 16 ) , ( 16 , 2 , 16 ) , ( 16 , 16 , 2 ) , ( 2 , 16 , 16 ) , ( 13 , 8 , 13 ) , ( 8 , 13 , 13 ) , ( 13 , 13 , 8 ) , ( 2 , 13 , 2 ) , ( 13 , 2 , 2 ) , ( 2 , 2 , 13 ) , ( 15 , 15 , 4 ) , ( 15 , 4 , 15 ) , ( 4 , 15 , 15 ) } .

For any prime number p ≡ 2 ( mod 3 ) , p ≡ 5 , 11 , 17 , or 23 ( mod 24 ) . It is obvious that the number ∣ S p ∣ , or N ( 1 , 3 , p ) , is a multiple of 3 since 3 ∣ ( p − 2 ) . We further claim that

Corollary 2.6

If p is an odd prime with p ≡ 2 ( mod 3 ) , then ∣ S p ∣ is divisible by 6 when p ≡ 5 , 17 , or 23 ( mod 24 ) . However, ∣ S p ∣ is divisible by 3 but not by 6 if p ≡ 11 ( mod 24 ) .

Proof

Theorem 2.4 gives

∣ S p ∣ = ( p − 1 ) ( p − 5 ) 8 , if p ≡ 5 or 17 ( mod 24 ) ; ( p − 1 ) ( p + 1 ) 8 , if p ≡ 11 or 23 ( mod 24 ) .

The proof follows immediately.□

Now we discuss the solution number N ( 3 , 2 , p ) , which is the number of non-trivial quadratic solutions to the equation x 3 + y 3 = 0 over Z / p Z . When p ≡ 2 ( mod 3 ) , N ( 3 , 2 , p ) is the same as the number N ( 1 , 2 , p ) of non-trivial quadratic solutions to the equation x + y = 0 and is also the number of non-trivial solutions to the equation x 6 + y 6 = 0 over Z / p Z . It is obvious that if − 1 is a quadratic non-residue mod p , then all of these equations have no solution, that is, N ( 3 , 2 , p ) = N ( 1 , 2 , p ) = 0 . However, if − 1 is a quadratic residue mod p , then ∀ a ∈ r p , ( a , − a ) is a non-trivial quadratic solution to the equation x 3 + y 3 = 0 and all of the non-trivial quadratic solutions to the equation x 3 + y 3 = 0 are of this form. Thus, N ( 3 , 2 , p ) = N ( 1 , 2 , p ) = p − 1 2 . By a similar proof as in the proof of Theorem 2.4, we also can obtain the same result. In summary,

Proposition 2.7

Let p be an odd prime with p ≡ 2 ( mod 3 ) . Then

N ( 3 , 2 , p ) = N ( 1 , 2 , p ) = 1 4 ( p − 1 ) 1 + ( − 1 ) p − 1 2 .

The situation for N ( 3 , 4 , p ) , the number of non-trivial solutions to x 6 + y 6 + z 6 + u 6 = 0 in Z / p Z , is similar. It can be evaluated as follows.

Theorem 2.8

If p is an odd prime with p ≡ 2 ( mod 3 ) , then

N ( 3 , 4 , p ) = N ( 1 , 4 , p ) = 1 16 ( p − 1 ) p 2 − 2 p + 3 + 6 ( − 1 ) p − 1 2 .

Proof

As before, the equation x 6 + y 6 + z 6 + u 6 = 0 is considered over the ring Z / p Z . Similarly as in the proof of Theorem 2.4,

N ( 3 , 4 , p ) = 1 16 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 ∑ d = 1 p − 1 a 3 + b 3 + c 3 + d 3 = 0 1 + a p 1 + b p 1 + c p 1 + d p = 1 16 ( p − 1 ) p 2 − 2 p + 3 + 6 ( − 1 ) p − 1 2 .

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 2 = x 5 − x is studied. It is shown that ∣ N ( p ) − p ∣ is relatively small compared to p . More precisely, ∣ N ( p ) − p ∣ < 4 p and the constant 4 is the best possible. In our study, all the non-trivial solutions to equation (1) are chosen from the set r p 3 whose size is ( p − 1 ) 3 / 8 . 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 .

Table 1

Density of solution number in r p 3

p 5 11 17 23 29 41
N ( 1 , 3 , p ) 0 15 24 66 84 180
∣ r p 3 ∣ 2 3 5 3 8 3 1 1 3 1 4 3 2 0 3
N ( 1 , 3 , p ) / ∣ r p 3 ∣ 0 0.12 0.046875 0.049587 0.030612245 0.0225

It seems when p is larger and larger, the density is smaller and smaller. We define the density function as follows.

Definition 2.9

Let p be a prime number. Define D ( p ) = N ( 1 , 3 , p ) ∣ r p 3 ∣ and call it the density function of the solutions to equation (1) in the set r p 3 .

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 1 / p . Note that, when p ≡ 2 ( mod 3 ) , p ≡ 5 , 11 , 17 , or 23 modulo 24.

Theorem 2.10

Let p be an odd prime number with p ≡ 2 ( mod 3 ) . Then

  1. N ( 1 , 3 , p ) ∣ r p 3 ∣ = p − 2 + 3 ( − 1 ) p + 1 2 ( p − 1 ) 2 .

  2. lim p → ∞ N ( 1 , 3 , p ) ∣ r p 3 ∣ = 0 and N ( 1 , 3 , p ) ∣ r p 3 ∣ = O ( 1 / p ) .

Proof

  1. From Theorem 2.4,

    D ( p ) = N ( 1 , 3 , p ) ∣ r p 3 ∣ = 1 8 ( p − 1 ) p − 2 + 3 ( − 1 ) p + 1 2 p − 1 2 3 = p − 2 + 3 ( − 1 ) p + 1 2 ( p − 1 ) 2 .

  2. Immediately from (1),

    lim p → ∞ N ( 1 , 3 , p ) ∣ r p 3 ∣ = lim p → ∞ p − 2 + 3 ( − 1 ) p + 1 2 ( p − 1 ) 2 = 0

    and

    lim p → ∞ N ( 1 , 3 , p ) ∣ r p 3 ∣ 1 / p = lim p → ∞ p − 2 + 3 ( − 1 ) p + 1 2 ( p − 1 ) 2 / p = 1 . □

It confirms that the density function D ( p ) is approaching to 0 when p goes to infinity. Actually, D ( p ) = O ( 1 / p ) .

3 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 ) . Theorem 2.4 states that the number of non-trivial solutions to the equation x 2 + y 2 + z 2 = 0 in Z / p Z , or the number of non-trivial quadratic solutions to x + y + z = 0 in Z / p Z , is N ( 1 , 3 , p ) and the set of such solutions is S p . Also,

∣ S p ∣ = N ( 1 , 3 , p ) = 1 8 ( p − 1 ) p − 2 + 3 ( − 1 ) p + 1 2 .

For a , b , c ∈ Z / p Z with a + b + c = 0 , we can view { a , b , c } as a partition of 0 in Z / p Z . Furthermore, if a , b , c are chosen from r p , we obtain a 3Q-partition of 0 in Z / p Z defined in 1.5. It is straightforward to check that there is no quadratic triple ( a , b , c ) ∈ r 5 3 satisfying a + b + c ≡ 0 ( mod 5 ) . Thus, S 5 = ∅ . The aforementioned formula also confirms that ∣ S 5 ∣ = N ( 3 , 3 , 5 ) = N ( 1 , 3 , 5 ) = 0 . For any prime p ≡ 2 ( mod 3 ) which is greater than 5, ∣ S p ∣ ≠ 0 . Next we consider primes p greater than 5 with p ≡ 2 ( mod 3 ) . For such a prime number p , S p ≠ ∅ and in Z / p Z , 3 a ≠ 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.

In Definition 1.5, the sets of all 3Q-partitions of 0 in r p , those of type I, and those of type II, are denoted as X ( p ) , X 1 ( p ) , X 2 ( p ) . Recall that for any ( a , b , c ) ∈ S p , a + b + c = 0 in Z / p Z and a , b , c ∈ r p . Thus, every triple in S p produces one 3Q partition of 0 in r p . On the other hand, for every 3Q-partition { a , b , c } of 0 in r p , that is, { a , b , c } ∈ X ( p ) , it gives three or six solution triples in S p , depending on ( a , b , c ) being of type I or II. If { a , a , c } ∈ X 1 ( p ) , where a ≠ c , a , c ∈ r p , and a + a + c = 0 , it produces three solutions of type I in S p : ( a , a , c ) , ( a , c , a ) , and ( c , a , a ) . Similarly, for { a , b , c } ∈ X 2 ( p ) , six solutions in S p of type II are produced by permutation. Recall that α p = ∣ X 1 ( p ) ∣ and β p = ∣ X 2 ( p ) ∣ . The relationships between α p , β p , and ∣ S p ∣ are given as follows:

Lemma 3.1

Let p be a prime with p ≡ 2 ( mod 3 ) . Then

∣ S p ∣ = 3 α p + 6 β p = 1 8 ( p − 1 ) p − 2 + 3 ( − 1 ) p + 1 2 .

Also note that, for any ( a , b , c ) ∈ S p , λ ( a , b , c ) ∈ S p as well for any λ ∈ r p . We define an equivalence relation on S p . One can easily check the equivalence property.

Definition 3.2

Let p be a prime number with p ≡ 2 ( mod 3 ) . For any two triples ( a , b , c ) , ( u , v , w ) ∈ S p , ( a , b , c ) ∼ ( u , v , w ) if and only if there exists λ ∈ r p such that ( a , b , c ) = ( λ u , λ v , λ w ) . The equivalence class containing ( x , y , z ) is denoted as [ ( a , b , c ) ] . The set of equivalence classes is denoted by S p ¯ .

Proposition 3.3

Let p be a prime number with p ≡ 2 ( mod 3 ) . Then

∣ S p ¯ ∣ = 1 4 p − 2 + 3 ( − 1 ) p + 1 2 and ∀ ( a , b , c ) ∈ S p , ∣ [ ( a , b , c ) ] ∣ = p − 1 2 .

Proof

Obviously, ∣ r p ∣ = ( p − 1 ) / 2 = ∣ [ ( a , b , c ) ] ∣ . The number of equivalence classes, ∣ S p ¯ ∣ , is given by ∣ S p ∣ / ( ( p − 1 ) / 2 ) .□

In this section, we provide formulas for α p , β p , and correspondingly the number of type I and type II solutions in S p . We first examine the situations for p = 29 and p = 41 .

Example 3.4

Consider p = 29 . The set of non-zero quadratic residues mod p is r 29 = { 1 , 4 , 5 , 6 , 7 , 9 , 13 , 16 , 20 , 22 , 23 , 24 , 25 , 28 } . From Proposition 3.3, ∣ S 29 ¯ ∣ = 6 . The six equivalence classes are as follows:

S 29 ¯ = { [ ( 1 , 4 , 24 ) ] , [ ( 4 , 1 , 24 ) ] , [ ( 1 , 24 , 4 ) ] , [ ( 24 , 1 , 4 ) ] , [ ( 24 , 4 , 1 ) ] , [ ( 4 , 24 , 1 ) ] } ,

which are all represented by a triple of type II. Thus, all the triples in S 29 are of type II. It implies that X 1 ( 29 ) = ∅ and so ∣ α 29 ∣ = 0 . The class [ ( 1 , 4 , 24 ) ] gives all the 14 partitions of 0 in r 29 : λ + 4 λ + 24 λ ≡ 0 ( mod 29 ) , where λ ∈ r 29 . Thus, X 2 ( 29 ) = { { λ , 4 λ , 14 λ } ∣ λ ∈ r 29 } and β 29 = ∣ r 29 ∣ = 14 . Consequently, the total number of solutions in S 29 is 6 ⋅ 14 = 84 , which confirms the result from Theorem 2.4: ∣ S 29 ∣ = N ( 1 , 3 , 29 ) = N ( 3 , 3 , 29 ) = 28 ( 27 − 3 ) / 8 = 84 .

For the prime number p = 41 , we obtain solutions of both types.

Example 3.5

For p = 41 , there are three equivalence classes of type I: [ ( 1 , 1 , 39 ) ] , [ ( 1 , 39 , 1 ) ] , [ ( 39 , 1 , 1 ) ] . There are six equivalence classes of type II: [ ( 1 , 4 , 36 ) ] , [ ( 1 , 36 , 4 ) ] , [ ( 4 , 36 , 1 ) ] , [ ( 4 , 1 , 36 ) ] , [ ( 36 , 1 , 4 ) ] , [ ( 36 , 4 , 1 ) ] .

In this example, ∣ r 41 ∣ = 20 , so each equivalence class has 20 triples from S 41 . Thus, S 41 has 3 ⋅ 20 = 60 solutions of type I and 6 ⋅ 20 = 120 solutions of type II. Altogether, ∣ S 41 ∣ = 60 + 120 = 180 . From Theorem 2.4, we also can calculate that ∣ S 41 ∣ = ( 40 / 8 ) ( 39 − 3 ) = 180 .

The equivalence classes [ ( 1 , 1 , 39 ) ] and [ ( 1 , 4 , 36 ) ] give the set of the partitions of 0 of type I and type II in r 41 , respectively. In particular,

X 1 ( 41 ) = { { λ , λ , 39 λ } ∣ λ ∈ r 41 } and X 2 ( 41 ) = { { λ , 4 λ , 36 λ } ∣ λ ∈ r 41 } .

So α 41 = β 41 = ∣ r 41 ∣ = 20 . Altogether, there are 40 3Q-partitions of 0 in r 41 , that is, ∣ X ( 41 ) ∣ = 40 . The size of S 41 can also be viewed as ∣ S 41 ∣ = 3 α 41 + 6 β 41 = 3 ⋅ 20 + 6 ⋅ 20 = 180 .

Proposition 3.6

Let p be a prime number greater than 5 and p ≡ 2 ( mod 3 ) .

  1. If p ≡ 5 or 23 ( mod 24 ) , then all the solutions in S p are of type II and ∣ S p ∣ is divisible by 6.

  2. If p ≡ 11 or 17 ( mod 24 ) , then there are exactly 3 ( p − 1 ) / 2 solutions of type I in S p .

  3. S 5 = ∅ and S 11 has no solutions of type II.

Proof

Let ( a , a , c ) ∈ S p , a solution triple of type I. Then a , c ∈ r p and a + a + c ≡ 0 ( mod p ) ⇒ − 2 a ≡ c ( mod p ) . It implies that − 2 is a quadratic residue mod p .

  1. Assume p ≡ 5 or 23 ( mod 24 ) . Then − 2 is not a quadratic residue of p . From the above, S p does not have solution triples of type I. Thus, all the solution triples in S p are of type II and α p = 0 . By Lemma 3.1, ∣ S p ∣ is divisible by 6.

  2. Assume p ≡ 11 or 17 ( mod 24 ) . Then − 2 is a quadratic residue mod p . Consider any solution triple ( a , a , c ) ∈ S p of type I as above. Since a ∈ r p , − 2 a ≡ k ( mod p ) for a unique k ∈ r p . It is obvious that all the three triples ( a , a , k ) , ( a , k , a ) , and ( k , a , a ) are solutions in S p of type I and there are exactly 3 ( p − 1 ) / 2 = 3 α p of them. Every solution triple of type I in S p must also be of such a form. Thus, S p has exactly 3 ( p − 1 ) / 2 many solutions of type I.

  3. From (2), S 11 has 3 ( 11 − 1 ) / 2 = 15 solutions of type I. By Lemma 3.1,

    ∣ S p ∣ = 1 8 ( 11 − 1 ) 11 − 2 + 3 ( − 1 ) 11 + 1 2 = 15 .

    Thus, S 11 has no solutions of type II.□

Next, we focus on 3Q-partitions of 0 in r p . We derive formulas α p , β p , and relationships between them. We examine an example first.

Example 3.7

Consider p = 11 . r 11 = { 1 , 4 , 9 , 5 = 4 2 , 3 = 5 2 } . From Proposition 3.6, X 2 ( 11 ) = ∅ .

X ( 11 ) = { { 1 , 1 , 9 } , { 1 , 5 , 5 } , { 3 , 3 , 5 } , { 3 , 4 , 4 } , { 4 , 9 , 9 } } = X 1 ( 11 ) .

Each above partition produces three solutions of type I in S 11 . The set S 11 is given as follows:

S 11 = { ( 1 , 1 , 9 ) , ( 1 , 9 , 1 ) , ( 9 , 1 , 1 ) , ( 1 , 5 , 5 ) , ( 5 , 1 , 5 ) , ( 5 , 5 , 1 ) , ( 3 , 3 , 5 ) , ( 3 , 5 , 3 ) , ( 5 , 3 , 3 ) , ( 3 , 4 , 4 ) , ( 4 , 3 , 4 ) , ( 4 , 4 , 3 ) , ( 4 , 9 , 9 ) , ( 9 , 4 , 9 ) , ( 9 , 9 , 4 ) } .

S 11 is divided into three equivalence classes: S 11 ¯ = { [ ( 1 , 1 , 9 ) ] , [ ( 1 , 9 , 1 ) ] , [ ( 9 , 1 , 1 ) ] } . Each of the equivalence class is of size 5. For example,

[ ( 1 , 1 , 9 ) ] = { ( 1 , 1 , 9 ) , 4 ( 1 , 1 , 9 ) = ( 4 , 4 , 3 ) , 9 ( 1 , 1 , 9 ) = ( 9 , 9 , 4 ) , 5 ( 1 , 1 , 9 ) = ( 5 , 5 , 1 ) , 3 ( 1 , 1 , 9 ) = ( 3 , 3 , 5 ) } .

We see that ∣ S 11 ∣ = ∣ S 11 ¯ ∣ ⋅ ∣ r 11 ∣ = 3 ⋅ 5 = 15 . Also, α 11 = 5 and β 11 = 0 .

We now give the formulas for α p , β p and show relationship between them.

Theorem 3.8

Let p be a prime number with p ≡ 2 ( mod 3 ) .

  1. If p ≡ 5 or 23 ( mod 24 ) , α p = 0 and

    β p = 1 48 ( p − 1 ) p − 2 + 3 ( − 1 ) p + 1 2 .

  2. If p ≡ 11 or 17 ( mod 24 ) , then α p = ( p − 1 ) / 2 , and

    β p = 1 48 ( p − 1 ) p − 14 + 3 ( − 1 ) p + 1 2 .

  3. The number ∣ X ( p ) ∣ = α p + β p of all the 3Q-partitions of 0 in r p is given as follows:

    ∣ X ( p ) ∣ = α p + β p = ( p − 1 ) ( p − 5 ) 48 , if p ≡ 5 ( mod 24 ) ; ( p − 1 ) ( p + 13 ) 48 , if p ≡ 11 ( mod 24 ) ; ( p − 1 ) ( p + 7 ) 48 , if p ≡ 17 ( mod 24 ) ; ( p − 1 ) ( p + 1 ) 48 , if p ≡ 23 ( mod 24 ) .

Proof

  1. If p ≡ 5 or 23 ( mod 24 ) , then there is no quadratic solution of type II by Proposition 3.6(1). Thus, α p = 0 . From Lemma 3.1,

    β p = ∣ S p ∣ 6 = ( p − 1 ) ( p − 2 + 3 ( − 1 ) ( p + 1 ) / 2 ) 48 .

  2. For p ≡ 11 or 17 ( mod 24 ) , there are 3 ( p − 1 ) / 2 many solutions of type I by Proposition 3.6(2). Since the three permutations from one solution provides one 3Q-partitions of 0, α p = ( p − 1 ) / 2 . Lemma 3.1 implies that

    β p = 1 6 ( ∣ S p ∣ − 3 α p ) = 1 6 1 8 ( p − 1 ) p − 2 + 3 ( − 1 ) p + 1 2 − 3 2 ( p − 1 ) = 1 48 ( p − 1 ) p − 14 + 3 ( − 1 ) p + 1 2 .

  3. Note that ( p + 1 ) / 2 is odd when p ≡ 5 or 17 ( mod 24 ) and is even when p ≡ 11 or 23 ( mod 24 ) . When p ≡ 5 or 23 ( mod 24 ) , α p = 0 . Then α p + β p = β p and the result follows from (1). For p ≡ 11 or 17 ( mod 24 ) , α p = p − 1 2 by Proposition 3.6(2). Add α p to β p obtained from (1) above, we have

    α p + β p = p − 1 2 + ( p − 1 ) p − 14 + 3 ( − 1 ) p + 1 2 48 = ( p − 1 ) ( p + 10 + 3 ( − 1 ) p + 1 2 ) 48 = ( p − 1 ) ( p + 13 ) 48 , if p ≡ 11 ( mod 24 ) ; ( p − 1 ) ( p + 7 ) 48 , if p ≡ 17 ( mod 24 ) . □

Corollary 3.9

Let p be a prime number with p ≡ 2 ( mod 3 ) .

  1. If β p = 0 , then p = 11 or p = 17 . That is, S p has no solutions of type II only when p = 11 or 17.

  2. If p ≡ 11 or 17 ( mod 24 ) and p ≠ 11 , p ≠ 17 , then

    24 p − 11 ≤ α p β p ≤ 24 p − 17 .

  3. If p ≡ 11 or 17 ( mod 24 ) and p ≠ 11 , p ≠ 17 , then 12 β p < α p 2 < 24 β p .

Proof

For (1), from Theorem 3.8, the only way to make β p = 0 is when p − 14 + 3 ( − 1 ) ( p + 1 ) / 2 = 0 . This only can occur when p = 11 or p = 17 .

For (2), p ≡ 11 or 17 ( mod 24 ) . We apply Theorem 3.8 (2) and Proposition 3.6 (2) to obtain

α p β p = p − 1 2 ( p − 1 ) ( p − 14 + 3 ( − 1 ) ( p + 1 ) / 2 ) 48 = 24 p − 14 + 3 ( − 1 ) ( p + 1 ) / 2 .

Since p − 17 ≤ p − 14 + 3 ( − 1 ) ( p + 1 ) / 2 ≤ p − 11 , we obtain the required result.

Similar to the proof of (2), for (3), assume p ≡ 11 or 17 ( mod 24 ) , p ≠ 11 , and p ≠ 17 . It implies that p > 33 . We have

α p 2 β p = ( p − 1 ) 2 4 ( p − 1 ) ( p − 14 + 3 ( − 1 ) ( p + 1 ) / 2 ) 48 = 12 ( p − 1 ) p − 14 + 3 ( − 1 ) ( p + 1 ) / 2 .

Because p > 33 , p − 1 < 2 ( p − 17 ) , thus 12 ( p − 1 ) / ( p − 17 ) < 24 . Then

12 = 12 ( p − 11 ) p − 11 < 12 ( p − 1 ) p − 11 ≤ 12 ( p − 1 ) p − 14 + 3 ( − 1 ) ( p + 1 ) / 2 ≤ 12 ( p − 1 ) p − 17 < 24 . □

Corollary 3.9(1) claims two cases and the only two cases when type II solution does not exist. It happens only when p = 11 or p = 17 . Example 3.7 confirms the truth when p = 11 . For p = 17 , we use Theorem 2.4 to calculate ∣ S 17 ∣ = 24 . S 17 has three equivalence classes each of which is of size 8: [ ( 1 , 1 , 15 ) ] , [ ( 1 , 15 , 1 ) ] , [ ( 15 , 1 , 1 ) ] . They produce all the 24 solutions in S 17 and they are all of type I which confirms that there is no solution of type II in S 17 .

Corollary 3.9(2) estimated the ratio α p / β p when α p ≠ 0 . However, the value of β p is closer to the value of α p 2 when α p ≠ 0 . Corollary 3.9(3) provides a comparison of these two values.

4 Counting monomials in three variables of certain degree in Z / p Z [ x , y , z ]

Polynomials are important functions which have many interesting properties and they play important roles in many areas of mathematics and science. For example, polynomials are used to approximate other complex functions. In advanced mathematics, polynomial rings and algebraic varieties over a field are central concepts in algebra and algebraic geometry. Among the polynomials with multiple variables, homogeneous polynomials are those with all the monomials having the same degree. Homogeneous polynomials are building blocks of the multivariable polynomials and they often appear in physics as a consequence of dimensional analysis, where measured quantities must match in real-world problems. Any non-zero polynomial can be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees. Consider a polynomial ring R = K [ x 1 , x 2 , … , x n ] over a field K and a positive integer d . It is known that the set R d of all homogeneous polynomials of degree d forms a vector space whose dimension is the number of different monomials of degree d (see [20]). We are interested in applying the results from the previous sections to calculate the number of certain monomials of the same degree d , for some values of d .

Consider the polynomial ring R = ( Z / p Z ) [ x , y , z ] with p being prime and a monomial x a y b z c ∈ R , where a , b , c are non-negative integers. By Fermat’s little theorem, r p = r for all r ∈ Z / p Z . Thus, we are only interested in the monomials x a y b z c with 0 ≤ a , b , c ≤ p − 1 . When a + b + c = d , we say x a y b z c is of degree d . For any non-constant monomial x a y b z c of degree d with 0 ≤ a , b , c ≤ p − 1 (or a , b , c ∈ Z / p Z ), 0 < d ≤ 3 ( p − 1 ) . We write x a y b z c = x α where x = ( x , y , z ) and α = ( a , b , c ) . A polynomial f ( x ) = f ( x , y , z ) ∈ R = ( Z / p Z ) [ x , y , z ] has the form of

f ( x ) = ∑ α ∈ T a α x α ,

where T is a subset of ( Z / p Z ) 3 and a α ∈ Z / p Z .

Definition 4.1

Consider the polynomial ring R = ( Z / p Z ) [ x , y , z ] , where p is odd and p ≡ 2 ( mod 3 ) . Let d be a non-negative integer. A polynomial in R is called a homogeneous polynomial of degree d if the degrees of all the involved monomials are of the same degree d .

Consider α = ( a , b , c ) , where a , b , c ∈ r p and a + b + c ≡ 0 ( mod p ) , that is, ( a , b , c ) ∈ S p . It implies that a + b + c = p or a + b + c = 2 p . In this section, we count the number of monomials of degree p or 2 p .

Definition 4.2

Define

S 1 = { ( a , b , c ) ∈ S p ∣ a + b + c = p } ; S 2 = { ( a , b , c ) ∈ S p ∣ a + b + c = 2 p } .

The number N ( 1 , 3 , p ) is the number of monomials x a y b z c of degree p or 2 p , where ( a , b , c ) ∈ S p = S 1 ∪ S 2 , a disjoint union. Then N ( 1 , 3 , p ) = ∣ S p ∣ = ∣ S 1 ∣ + ∣ S 2 ∣ . The values of ∣ S p ∣ , ∣ S 1 ∣ , and ∣ S 2 ∣ for small primes are indicated in Table 2.

Table 2

Values of ∣ S p ∣ , ∣ S 1 ∣ , and ∣ S 2 ∣

p 11 17 23 29 41 47 53 59 71 83 89 101 ⋯
∣ S p ∣ 15 24 66 84 180 276 312 435 630 861 924 1,200 ⋯
∣ S 1 ∣ 12 12 60 42 90 228 156 285 504 525 462 600 ⋯
∣ S 2 ∣ 3 12 6 42 90 48 156 150 126 336 462 600 ⋯

We note that in Table 2, ∣ S 1 ∣ = ∣ S 2 ∣ for some primes such as 17, 29, 41, 53, and so on. We classify the prime numbers achieving this property as follows.

Theorem 4.3

Let p be a prime number with p ≡ 2 ( mod 3 ) .

  1. There are exactly 1 8 ( p − 1 ) p − 2 + 3 ( − 1 ) p + 1 2 many monomials of degree p or 2 p ;

  2. If p ≡ 5 or 17 ( mod 24 ) , then

    ∣ S 1 ∣ = ∣ S 2 ∣ = 1 2 N ( 1 , 3 , p ) = 1 16 ( p − 1 ) ( p − 5 ) .

    That is, there are exactly ( p − 1 ) ( p − 5 ) / 16 many monomials of degree d and so does that of degree 2 p .

Proof

  1. It is straightforward from Theorem 2.4 which gives the value of ∣ S p ∣ .

  2. If p ≡ 5 or 17 ( mod 24 ) , then − 1 is a quadratic residue mod p . Thus, a + b + c = p ⇔ ( p − a ) + ( p − b ) + ( p − c ) = 2 p . Here 0 < a , b , c < p ⇔ 0 < ( p − a ) , ( p − b ) , ( p − c ) < p . Thus, there is a one-to-one and onto correspondence between S 1 and S 2 . Then apply Theorem 2.4 again, ∣ S 1 ∣ = ∣ S 2 ∣ = 1 2 N ( 1 , 3 , p ) .□

Note that for any monomial x α , where α ∈ S i , the exponent for each of the variables x , y , z is non-zero quadratic residue mod p and the total degree is p for i = 1 and 2 p for i = 2 . The aforementioned results provided the number of such monomials. Next, we show that these types of monomials have an interesting formal partial derivative. Let x α be a monomial in the polynomial ring ( Z / p Z ) [ x 1 , … , x n ] , the formal partial derivative of x α with respect to x i is ∂ x α ∂ x i .

Theorem 4.4

Let p be an odd prime number with p ≡ 2 ( mod 3 ) and R = ( Z / p Z ) [ x , y , z ] .

  1. Let x α be a monomial in R with α ∈ S p . Then

    x ∂ x α ∂ x + y ∂ x α ∂ y + z ∂ x α ∂ z = 0 .

  2. A polynomial f ( x ) in R satisfying x ∂ f ∂ x + y ∂ f ∂ y + z ∂ f ∂ z = 0 if all of the monomial terms of f ( x , y , z ) are of degree p or 2 p .

Proof

  1. The left hand side expression equals 0 if and only if the coefficients of all the three monomials are divisible by p . Let x α = x a y b z c , where ( a , b , c ) ∈ S p . Then a + b + c = p or a + b + c = 2 p . Then

    x ∂ x α ∂ x + y ∂ x α ∂ y + z ∂ x α ∂ z = x ( a x a − 1 y b z c ) + y ( b x a y b − 1 z c ) + z ( c x a y b z c − 1 ) = ( a + b + c ) x a y b z c = 0 in R ,

    since a + b + c = p or 2 p which means a + b + c = 0 in R .

  2. This is immediately from (1).□

5 Conclusions and future directions

In this paper, we have made connections between the number of solutions of certain Diophantine equations modulo a prime number p and the number of 3Q-partitions of 0 in the ring Z / p Z . We classified two types of 3Q-partitions of 0 and provided the number of each type of partitions. Many questions remain open. In the theory of partitions of positive integers, researchers have worked on restricted partitions with a given number of parts, with distinct parts, or with a given maximal value of all the parts, etc. Other questions involve concepts such as conjugate and self-conjugate partitions. We are interested in studying similar problems for our partition problems. We also plan to study similar Diophantine equations mod p involving more than three variables and generalize the results.

In Section 4, we counted the number of certain monomials of degree p or 2 p in the polynomial ring ( Z / p Z ) [ x , y , z ] . We provided exact formulas for the number of each type of monomials when p ≡ 5 or 17 ( mod 24 ) . It remains open to count such monomials for p ≡ 11 or 23 ( mod 24 ) . Let R p = { all monomials of degree p } and R 2 p = { all monomials of degree 2 p } . It is known from [20] that R p and R 2 p are vector spaces over Z / p Z and the dimensions of these two spaces are given by

dim ( R p ) = p + 2 2 = ( p + 2 ) ( p + 1 ) 2 , dim ( R 2 p ) = 2 p + 2 2 = ( p + 1 ) ( 2 p + 1 ) .

The set of monomials we have studied is a subset of R p ∪ R 2 p . It is natural to study other types of monomials and the enumeration of them.

A homogeneous ideal of the ring ( Z / p Z ) [ x 1 , … , x n ] is an ideal generated by homogeneous polynomials. These ideals are important algebraic structures in the study of projective varieties in algebraic geometry. We would like to seek further connections and applications in this area.

Acknowledgement

The authors would like to thank the referees for helpful and detailed comments, which have significantly contributed to improving the presentation of this paper.

  1. Funding information: This work was supported by the Fundamental Research Funds for the Central Universities, CHD (300102121102).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-09-15
Revised: 2020-12-19
Accepted: 2021-02-04
Published Online: 2021-05-03

© 2021 Wang Xiao and Aihua Li, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0032
Keywords for this article
partition of a number; Dirichlet character sum; Diophantine equation; solution number
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  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
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  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
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  89. Global optimization and applications to a variational inequality problem
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  93. Compact perturbations of operators with property (t)
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  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
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  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
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  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
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  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
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  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
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  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
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  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
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  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
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