IJMMS 2003:19, 1175–1183
PII. S0161171203206256
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON THE SPECIAL SOLUTIONS OF AN EQUATION
IN A FINITE FIELD
LI HAILONG and ZHANG WENPENG
Received 10 June 2002
The main purpose of this paper is to prove the following conclusion: let p be a
prime large enough and let k be a fixed positive integer with 2k|p − 1. Then for
any finite field Fp and any element 0 ≠ c ∈ Fp , there exist three generators x, y,
and z ∈ Fp such that x k y k + y k zk + x k zk = c.
2000 Mathematics Subject Classification: 11E12, 11T23.
1. Introduction. Let p be an odd prime, let k be a fixed positive integer
with 2k|p − 1, and let Fp be the finite field with p elements. It is clear that
there exists at least one generator of Fp , and the number of all generators of
Fp is equal to φ(p − 1), where φ(n) is Euler’s function. The main purpose of
this paper is to study the following two problems:
(A) for any element 0 ≠ c ∈ Fp whether there exist three generators x, y,
and z ∈ Fp such that
x k y k + y k zk + x k zk = c;
(1.1)
(B) if (A) is true, let N(c, k, p) denotes the number of all solutions of (1.1).
What can be said about the asymptotic properties of N(c, k, p)?
In this paper, we use the estimates for general Gauss sums and the properties of Dirichlet characters to study the above two problems and prove the
following main conclusion.
Theorem 1.1. Let p be an odd prime and k a fixed positive integer with
2k|p − 1. Then for any element 0 ≠ c ∈ Fp , the asymptotic formula
N(c, k, p) =
φ3 (p − 1)
φ3 (p − 1)
+θ ·
· p · 8ω(p−1) ,
p
(p − 1)3
(1.2)
where |θ| ≤ 54k4 and ω(n) denotes the number of all distinct prime divisors
of n.
1176
L. HAILONG AND Z. WENPENG
From this theorem, we may immediately deduce the following corollary.
Corollary 1.2. Let p be a prime large enough and k a fixed positive integer
with 2k|p −1. Then for any integer 1 ≤ c ≤ p −1, there exist three primitive roots
x, y, and z modulo p such that the congruence
x k y k + y k zk + x k zk ≡ c mod p.
(1.3)
2. Some lemmas. In this section, we give several lemmas which are necessary in the proof of Theorem 1.1. First, we let
nak
χ(a)e
G(n, χ, k; q) = G(n, χ; q) =
,
q
a=1
q
(2.1)
where χ denotes a Dirichlet character mod q, e(y) = e2π iy . Then we have the
following lemma.
Lemma 2.1. Let p be an odd prime and k a positive integer with k|p − 1.
Then for any integer n with p n,

√
≤ k p,
G(n, χ, k; p)
= 0,
if χ (p−1)/k = χ0 ,
otherwise,
(2.2)
where χ0 denotes the principal character mod p.
Proof. Let g be a fixed primitive root mod p, then for any integer n with
p n, there exist two integers l and i such that n ≡ g lk+i (mod p), here 0 ≤
i < k. If b runs through a complete residue system mod p, then g l b also runs
through a complete residue system mod p, so that we have
p
p
g lk+i ak − bk
G(n, χ, k; p)2 =
χ ab e
p
a=1 b=1
p
p
k
g i g l b ak − 1
χ(a)e
=
p
a=1 b=1
p
p
g i bk ak − 1
=
.
χ(a)
e
p
a=1
b=1
(2.3)
Note the trigonometric identity


q
q,
ma
=
e

q
0,
a=1
if q|m,
if q m.
(2.4)
ON THE SPECIAL SOLUTIONS OF AN EQUATION IN A FINITE FIELD
1177
Further, g i bk , i = 0, . . . , k − 1; b = 1, . . . , p runs through k complete residue
systems mod p so that we have the identity
p
p k−1 i k k
i
g b a −1
G g , χ, k; p 2 =
χ(a)
e
p
a=1
i=0
b=1 i=0
k−1
b ak − 1
χ(a)
e
=k
p
a=1
b=1
p
p−1
p−1
= kp
χ(a)
(2.5)
a=1
ak ≡1(mod p)
= kp 1 + χ g (p−1)/k + · · · + χ g (k−1)(p−1)/k
=


k2 p,
if χ (p−1)/k = χ0 ,

0,
otherwise.
From (2.5), we easily get the estimate
1/2

k−1
2
G g i , χ, k; p 
G(n, χ, k; p) ≤ 
i=0
 √
k p, if χ (p−1)/k = χ0 ,
=
0,
otherwise.
(2.6)
This proves Lemma 2.1.
Lemma 2.2. Let p be an odd prime and let n be an integer. Then

 p −1
k

µ(k) ,
a ind n
= φ(p − 1)
e

φ(k) a=1
k
0,
k|p−1
if n is a primitive root of p,
(2.7)
otherwise,
where ind n denotes the index of n relative to some fixed primitive root of p,
k
µ(n) is the Möbius function, and a=1 denotes the summation over all a such
that (a, k) = 1.
Proof. See [1, Proposition 2.2].
Lemma 2.3. Let p be an odd prime, let k a fixed positive integer with 2k|p −
1, and let χ1 , χ2 , and χ3 be three Dirichlet characters mod p. Then for any
1178
L. HAILONG AND Z. WENPENG
integer u with (u, p) = 1,
p−1
p−1
p−1
u ak bk + bk c k + c k ak ≤ 8k3 p p.
χ
(a)χ
(b)χ
(c)e
1
2
3
p
a=1 b=1 c=1
(2.8)
Proof. Let g be any fixed primitive root mod p, then, from the properties
of primitive roots and reduced residue system mod p, we have
u ak bk + bk c k + c k ak
χ1 (a)χ2 (b)χ3 (c)e
p
a=1 b=1 c=1
p−1 p−1 p−1
u ak bk c k + bk c k + c 2k ak
=
χ1 (c)χ1 (a)χ2 (b)χ3 (c)e
p
a=1 b=1 c=1
p−1 p−1 p−1
u ak bk + bk + c 2k ak
(2.9)
=
χ1 (c)χ1 (a)χ2 (c)χ2 (b)χ3 (c)e
p
a=1 b=1 c=1
p−1
ubk
=
χ2 (b)e
p
b=1
p−1
p−1
uak bk uak c 2k
×
.
χ1 (a)e
χ1 (c)χ2 (c)χ3 (c)e
p
p
a=1
c=1
p−1 p−1 p−1
Let h be a fixed quadratic nonresidue modulo p, then
p−1
χ1 (a)e
a=1
uak bk
p
p−1
χ1 (c)χ2 (c)χ3 (c)e
c=1
uak c 2k
p
2 t us 2k hkt bk
χ1 s h e
p
s=1 t=0
p−1
us 2k hkt c 2k
×
χ1 (c)χ2 (c)χ3 (c)e
p
c=1
1
=
2
p−1 1
p−1
1
uhkt s 2k bk
1 t 2
χ1 h
χ1 s χ1 (s)χ2 (s)χ3 (s)e
=
2 t=0
p
s=1
p−1
uhkt c 2k
χ1 (c)χ2 (c)χ3 (c)e
×
p
c=1
=
1
1 t χ1 h G uhkt , χ1 χ2 χ3 , 2k; p
2 t=0
p−1
uhkt s 2k bk
×
.
χ1 (s)χ2 (s)χ3 (s)e
p
s=1
(2.10)
ON THE SPECIAL SOLUTIONS OF AN EQUATION IN A FINITE FIELD
1179
From (2.9) and (2.10), we have
p−1 p−1 p−1
χ1 (a)χ2 (b)χ3 (c)e
a=1 b=1 c=1
=
=
=
u ak bk + bk c k + c k ak
p
1
1 t χ1 h G uhkt , χ1 χ2 χ3 , 2k; p
2 t=0
p−1
p−1
ubk uhkt a2k bk
×
χ2 (b)e
χ1 (a)χ2 (a)χ3 (a)e
p
p
a=1
b=1
1
1 t χ1 h G uhkt , χ1 χ2 χ3 , 2k; p
4 t=0
p−1 1
us 2k hr k
×
χ2 s 2 hr e
p
s=1 r =0
p−1
uhkt s 2k hr k a2k
χ1 (a)χ2 (a)χ3 (a)e
×
p
a=1
(2.11)
1
1
1 t r χ1 h χ2 h G uhkt , χ1 χ2 χ3 , 2k; p
4 r =0 t=0
× G uhr k , χ1 χ2 χ3 , 2k; p G uhkt hr k , χ1 χ2 χ3 , 2k; p .
Applying Lemma 2.1 to (2.11), we immediately get the estimate
p−1
p−1
p−1
u ak bk + bk c k + c k ak ≤ 8k3 p p.
χ1 (a)χ2 (b)χ3 (c)e
p
a=1 b=1 c=1
(2.12)
This proves Lemma 2.3.
3. Proof of the Theorem 1.1. We only prove that Theorem 1.1 is true if Fp is
a complete residue system modulo p, then, from the isomorphism properties
of the finite field, we can deduce that Theorem 1.1 is true for any finite field
Fp . Let p be an odd prime and Ꮽ(p) = Ꮽ denotes the set of all primitive roots
modulo p in the interval [1, p − 1], then, from the trigonometric identity (2.4)
and Lemma 2.2, we have
N(c, k, p) =
1
u∈Ꮽ v∈Ꮽ w∈Ꮽ
uk v k +v k w k +uk w k ≡c(p)
=
φ3 (p − 1) µ(j) µ(h) µ(l)
p(p − 1)3 j|p−1 h|p−1 l|p−1 φ(j) φ(h) φ(l)
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L. HAILONG AND Z. WENPENG
j
l p−1 p−1 p−1
h
a ind u b ind v d ind w
+
+
×
e
j
h
l
a=1 b=1 d=1 u=1 v=1 w=1
p
t uk v k + v k w k + w k uk − c
e
×
p
t=1
φ3 (p − 1) µ(j) µ(h) µ(l)
p(p − 1)3 j|p−1 h|p−1 l|p−1 φ(j) φ(h) φ(l)
=
j
h
l p−1 p−1 p−1
a ind u b ind v d ind w
+
+
e
×
j
h
l
a=1 b=1 d=1 u=1 v=1 w=1
+
φ3 (p − 1) µ(j) µ(h) µ(l)
p(p − 1)3 j|p−1 h|p−1 l|p−1 φ(j) φ(h) φ(l)
j
h
l p−1 p−1 p−1
a ind u b ind v d ind w
+
+
×
e
j
h
l
a=1 b=1 d=1 u=1 v=1 w=1
p−1
t uk v k + v k w k + w k uk − c
e
×
p
t=1
≡
φ3 (p − 1) R1 + R2 .
p(p − 1)3
(3.1)
p−1
First, we estimate the main term R1 . Note (2.4) and a=1 χ(a) = 0 (χ is a nonprincipal character modulo p), from the definition of Dirichlet characters, we
have
j p−1 µ(j) a ind u
e
R1 = (p − 1) + 3(p − 1)
φ(j) a=1 u=1
j
j|p−1
3
2
j>1
j
h p−1 p−1 µ(j) µ(h) a ind u b ind v
+ 3(p − 1)
+
e
φ(j) φ(h) a=1 b=1 u=1 v=1
j
h
j|p−1 h|p−1
j>1
h>1
µ(j) µ(h) µ(l)
+
φ(j) φ(h) φ(l)
j|p−1 h|p−1 l|p−1
h>1
j>1
j
h
l>1
l
p−1
p−1
p−1
a ind u b ind v d ind w
+
+
e
×
j
h
l
a=1 b=1 d=1 u=1 v=1 w=1
= (p − 1)3 + 3(p − 1)2
j p−1
µ(j) χ(u; a, j)
φ(j) a=1 u=1
j|p−1
j>1
+ 3(p − 1)
j
h p−1
p−1
µ(j) µ(h) χ(v; b, h)
χ(u; a, j)
φ(j) φ(h) a=1 b=1 v=1
u=1
j|p−1 h|p−1
j>1
h>1
ON THE SPECIAL SOLUTIONS OF AN EQUATION IN A FINITE FIELD
j>1
h>1
1181
µ(j) µ(h) µ(l)
+
φ(j) φ(h) φ(l)
j|p−1 h|p−1 l|p−1
×
l>1
j
l p−1
h
p−1
χ(v; b, h)
a=1 b=1 d=1 v=1
p−1
χ(u; a, j)
u=1
χ(w; d, l),
w=1
(3.2)
where χ(u; a, j) = e(a ind u/j), χ(v; b, h) = e(b ind v/h), and χ(w; d, l) =
e(d ind w/l) are three Dirichlet characters mod p. Since j > 1, h > 1, l > 1, and
(b, h) = (a, j) = (d, l) = 1, the characters χ(u; a, j), χ(v; b, h), and χ(w; d, l)
are three primitive characters mod p. Therefore, we have
p−1
p−1
χ(u; a, j) =
u=1
p−1
χ(v; b, h) =
v=1
χ(w; d, l) = 0.
(3.3)
w=1
From these identities and (3.2), we immediately get the main term
R1 = (p − 1)3 .
(3.4)
In order to estimate the error term R2 in (3.1), first we separate R2 into four
parts. That is,
t uk v k + v k w k + w k uk − c
e
R2 =
p
u=1 v=1 w=1 t=1
p−1 p−1 p−1 p−1
j p−1 p−1 p−1 µ(j) a ind u
e
+3
φ(j) a=1 u=1 v=1 w=1
j
j|p−1
j>1
t uk v k + v k w k + w k u k − c
×
e
p
t=1
p−1
+3
j
h p−1 p−1 p−1 µ(j) µ(h) a ind u b ind v
+
e
φ(j) φ(h) a=1 b=1 u=1 v=1 w=1
j
h
j|p−1 h|p−1
j>1
h>1
p−1 k k
t u v + v k w k + w k uk − c
×
e
p
t=1
+
µ(j) µ(h) µ(l)
φ(j) φ(h) φ(l)
j|p−1 h|p−1 l|p−1
j>1
h>1
l>1
1182
L. HAILONG AND Z. WENPENG
j
l p−1 p−1 p−1
h
a ind u b ind v d ind w
+
×
e
j
h
l
a=1 b=1 d=1 u=1 v=1 w=1
p−1
t uk v k + v k w k + w k u k − c
e
×
p
t=1
≡ R21 + R22 + R23 + R24 .
(3.5)
Let g be any fixed primitive root mod p. Then note that 2k|p − 1, from the
properties of primitive roots and reduced residue system mod p, we have
t uk v k + v k w k + w k uk − c
p
u=1 v=1 w=1 t=1
p−2 2k−1 p−1 p−1 p−1 2ks+t k k
g
u v + v k w k + w k uk − c
1 e
=
2k s=0 t=0 u=1 v=1 w=1
p
p−1 p−1 p−1 p−2 2k−1
g t g 2ks uk v k + v k w k + w k uk
−cg t g 2ks 1 e
e
=
2k t=0 s=0
p
p
u=1 v=1 w=1
p−1 p−1 p−1 p−1 2k−1
g t uk v k + v k w k + w k uk
−cg t a2k 1 .
e
e
=
2k t=0 a=1
p
p
u=1 v=1 w=1
p−1 p−1 p−1 p−1
R21 =
e
(3.6)
Applying Lemmas 2.1 and 2.3 to (3.6), we immediately get
R21 ≤ 16k4 p 2 .
(3.7)
Using the same method of proving (3.7) and Lemma 2.3, and noting the identity
ω(n)
, we can also get
d|n |µ(d)| = 2
p−1 p−1
p−1
p−1
t uk v k + v k w k + w k uk − c a ind u ≤ 16k4 p 2 (3.8)
e
e
j
p
u=1 v=1 w=1
t=1
or
R23 ≤ 48k4 p 2 · 4ω(p−1) ,
R22 ≤ 48k4 p 2 · 2ω(p−1) ,
R24 ≤ 16k4 p 2 · 8ω(p−1) .
(3.9)
From (3.5), (3.7), and (3.9), and noting that ω(p − 1) ≥ 1, we get
R2 ≤ 54k4 p 2 · 8ω(p−1) .
(3.10)
ON THE SPECIAL SOLUTIONS OF AN EQUATION IN A FINITE FIELD
1183
Combining (3.1), (3.4), and (3.10), we obtain
N(c, k, p) =
φ3 (p − 1)
φ3 (p − 1)
+θ ·
· p · 8ω(p−1) ,
p
(p − 1)3
(3.11)
where |θ| ≤ 54k4 . This completes the proof of Theorem 1.1.
Note 3.1. Using the similar method of proving Theorem 1.1, we can also
get the asymptotic formula
N(0, k, p) =
φ3 (p − 1)
φ3 (p − 1) + θ1 ·
· p · 4ω(p−1) ,
p
(p − 1)2
(3.12)
where |θ1 | ≤ k3 .
Acknowledgment. This work was supported by the National Science
Foundation (NSF) and Province Natural Science Foundation (PNSF) of China.
References
[1]
W. Narkiewicz, Classical Problems in Number Theory, Monografie Matematyczne,
vol. 62, Państwowe Wydawnictwo Naukowe (PWN-Polish Scientific Publishers), Warsaw, 1986.
Li Hailong: Department of Mathematics, Weinan Teacher’s College, Weinan, Shaanxi,
China
E-mail address: lihailong@163.com
Zhang Wenpeng: Research Center for Basic Science, Xi’an Jiaotong University, Xi’an,
Shaanxi, China
E-mail address: wpzhang@nwu.edu.cn