Let \(q\) be a positive integral power of some prime \(p\) and \(\mathbb{F}_{q^m}\) be a finite field with \(q^m\) elements for some \(m \in \mathbb{N}\). Here we establish a sufficient condition for the existence of primitive normal pairs of the type \((\epsilon, f(\epsilon))\) in \(\mathbb{F}_{q^m}\) over \(\mathbb{F}_{q}\) with two prescribed traces, \(\text{Tr}_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\) and \(\text{Tr}_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\), where \(f(x) \in \mathbb{F}_{q^m}(x)\) is a rational function with some restrictions and \(a, b \in \mathbb{F}^*_q\). Furthermore, for \(q=5^k\), \(m \geq 9\) and rational functions with degree sum 4, we explicitly find at most 13 fields in which the desired pair may not exist.
Given a prime power \(q\) and a positive integer \(m\), we denote the finite field with \(q\) elements by \(\mathbb{F}_q\) and its extension field of degree \(m\) by \(\mathbb{F}_{q^m}\). The multiplicative group \(\mathbb{F}^*_{q^m}\) is a cyclic group and a generator of this group is called a primitive element of \(\mathbb{F}_{q^m}\). For an element \(\epsilon\in\mathbb{F}_{q^m}\), the elements \(\epsilon,\epsilon^q, \epsilon^{q^2},\ldots, \epsilon^{q^{m-1}}\) are called conjugates of \(\epsilon\) with respect to \(\mathbb{F}_q\). Sum of all these conjugates is called the trace of \(\epsilon\) over \(\mathbb{F}_q\) and is denoted by \(T_{\mathbb{F}_{q^m}/\mathbb{F}_q}(\epsilon)\). Further, an element \(\epsilon\in\mathbb{F}_{q^m}\) for which the set \(\{\epsilon,\epsilon^q, \epsilon^{q^2},\ldots, \epsilon^{q^{m-1}}\}\) forms a basis of \(\mathbb{F}_{q^m}( \mathbb{F}_q)\), when considered as a vector space, is called a \(normal\) element. An element that is simultaneously primitive and normal is referred to as a primitive normal element. Existence of such elements is firstly given by the following theorem.
Theorem 1.1 (Primitive normal basis theorem).In the finite field \(\mathbb{F}_{q^m}\), there always exists some element which is simultaneously primitive and normal.
Lenstra and Schoof proved this result in [14]. Later, Cohen and Huczynska [10] presented a modified proof of this result with the help of sieving techniques.
In 1985, Cohen [8] proved the existence of consecutive pairs of primitive elements of the type \((\epsilon, \epsilon^{-1})\) in \(\mathbb{F}_q\). After that, it was further developed in [1, 9, 12], among others. Similar studies on primitive normal pairs emerged following the development of the primitive normal basis theorem; they can be studied from [3, 13, 17] etc. and the references therein.
In 2001, Chou and Cohen [5] studied existence of primitive pair \((\epsilon, \epsilon^{-1})\) in \(\mathbb{F}_q\) such that both have trace zero over \(\mathbb{F}_p\). Cao and Wang [2] proved that, for any prime power \(q\) and any positive integer \(m\), there exists an element \(\epsilon \in \mathbb{F}_{q^m}\) such that \(\epsilon+\epsilon^{-1}\) is a primitive element of \(\mathbb{F}_{q^m}\) and \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\), \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon^{-1})=b\) for any prescribed \(a, b \in \mathbb{F}^*_{q}\). In [6], Choudhary et al. discussed the existence of primitive pairs \((\epsilon, f(\epsilon))\) with two prescribed traces, where \(f\) is a rational function with some restrictions. Recently Sharma et al. [16] considered rational functions \(f\) with some minor restrictions over \(\mathbb{F}_{q^m}\) for some prime power \(q\) and \(m \in \mathbb{N}\) and provided a sufficient condition for the existence of primitive normal pair of the form \((\epsilon, f(\epsilon))\) over \(\mathbb{F}_q\).
Inspired by all these results, we are presenting a sufficient condition for the existence of primitive normal pair \((\epsilon,f(\epsilon))\) in \(\mathbb{F}_{q^m}\) over \(\mathbb{F}_q\) such that \(T_{\mathbb{F}_{q^m}/\mathbb{F}_q}(\epsilon)=a\) and \(T_{\mathbb{F}_{q^m}/\mathbb{F}_q}(f(\epsilon))=b\), where \(f(x) \in \mathbb{F}_{q^m}(x)\) is a rational function with some restrictions and \(a,b \in \mathbb{F}^*_q\). The kind of rational functions we will study in this paper are defined below:
Definition 1.2. The set \(\mathcal{R}_{q^m}^n\) consists of rational functions of the simplest form \(f=\frac{f_1}{f_2}\) of degree sum \(n=\deg(f_1)+\deg(f_2)\), such that
(i) \(f \neq cx^{j}h^{d}\) for any \(c \in \mathbb{F}^*_{q^m}\), any \(j \in \mathbb{Z}\), any \(h\in \mathbb{F}_{q^m}(x)\) and any divisor \(d\) other than \(1\) of \(q^m-1\).
(ii) There exists at least one monic irreducible factor \(g\) of \(f_2\) in \(\mathbb{F}_{q^m}[x]\) with multiplicity \(r\) such that \(q^m \nmid r\).
We will use the notations \(\deg(f_1)=n_1\) and \(\deg(f_2)=n_2\) throughout. We also note that the second condition of the above definition implies that \(n_2>0\). This type of rational function is popularly known as Cohen’s kind of rational function [12] which also includes Carvalho’s kind of rational function [3].
In Section 3, we derive a sufficient condition for the existence of desired primitive normal pairs. In Section 4, we use a sieving inequality to improve the sufficient condition for more efficient results. In Section 5, we study the application of our results over fields of characteristic \(5\) and for rational functions of degree sum \(4\) and prove the following:
Theorem 1.3. Let \(q=5^k, m\geq9\) and \(f \in \mathcal{R}_{q^m}^n\). Then, for any \(a,b \in \mathbb{F}^*_q\), there exists an element \(\epsilon \in \mathbb{F}_{q^m}\) with \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\), \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\) and \((\epsilon,f(\epsilon))\) is a primitive normal pair in \(\mathbb{F}_{q^m}\) over \(\mathbb{F}_q\) unless \((q,m)\) is one of the \(13\) pairs: \((5,9)\), \((5,10)\), \((5,11)\), \((5,12)\), \((5,14)\), \((5,16)\), \((5,18)\), \((5,20)\), \((5,24)\), \((5^2,9)\), \((5^2,10)\), \((5^2,12)\), \((5^2,24)\).
Now, we present some notations that are significant to this article. We indicate the algebraic closure of \(\mathbb{F}_{q}\) by \(\mathbb{F}\). Let \(\mathcal{A}_p^n\) be the set of pairs \((q,m)\) such that there exists an element \(\epsilon \in \mathbb{F}_{q^m}\) with \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\) and \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\), where \(f(x) \in \mathcal{R}_{q^m}^n\) and \(a, b \in \mathbb{F}^*_q\). For each \(n \in \mathbb{N}\), we denote the number of prime divisors of \(n\) and the number of square-free divisors of \(n\) by \(w(n)\) and \(W(n)\), respectively. Additionally, for \(r(x) \in \mathbb{F}_q[x]\), we indicate the number of square-free and monic irreducible \(\mathbb{F}_q\)-divisors of \(r\) by \(W(r)\) and \(w(r)\), respectively.
In this section, we review a few definitions, lemmas and notations that will be used in proving a sufficient condition for the existence of primitive normal pairs of our interest.
Definition 2.1 (Characters).Let \(G\) be a finite abelian group. Then, a character \(\chi\) of \(G\) is a homomorphism from \(G\) to the group \(S^1:= \{z \in \mathbb{C} : |z| = 1\}\). The characters of \(G\) form a group under multiplication called the \(\mathit{dual\thinspace group}\) or \(\mathit{character\thinspace group}\) of \(G\), which is denoted by \(\widehat{G}\). It is well known that \(\widehat{G}\) is isomorphic to \(G\). Also, \(\chi_1\) denotes the trivial character of \(G\) defined as \(\chi_1(a)=1\), for all \(a \in G\).
A character of a group \(G\) of order \(d\) is denoted by \(\chi_d\). We also denote all the characters of order \(d\) by \((d)\).
Corresponding to two different abelian group structures present in a finite field \(\mathbb{F}_{q^m}\) i.e., \((\mathbb{F}_{q^m}, +)\) and \((\mathbb{F}^*_{q^m},\cdot)\); two types of characters can be formed. We refer to them as additive character denoted by \(\psi\) and multiplicative character denoted by \(\chi\), respectively. Multiplicative characters are extended from \(\mathbb{F}^*_{q^m}\) to \(\mathbb{F}_{q^m}\) by the rule \(\chi(0)=\begin{cases} 0 \,\mbox{ if}\, \chi\neq\chi_1,\\ 1 \,\mbox{ if}\, \chi=\chi_1. \end{cases}\)
The additive character \(\bar{\psi}(\epsilon)=e^{2{\pi}iT(\epsilon)/p}\), for all \(\epsilon \in \mathbb{F}_{q}\), where \(T\) is the absolute trace function from \(\mathbb{F}_{q}\) to \(\mathbb{F}_p\), is called the canonical additive character of \(\mathbb{F}_{q}\) and every additive character \(\psi_{\alpha}\) for \(\alpha \in \mathbb{F}_{q}\) can be expressed in terms of the canonical additive character \(\bar{\psi}\) as \(\psi_{\alpha}(\beta)=\bar{\psi}(\alpha\beta)\) for all \(\beta \in \mathbb{F}_{q}\).
Definition 2.2. (\(e\)-free element) For any divisor \(e\) of \(q^m-1\), an element \(\epsilon \in \mathbb{F}_{q^m}\) is called \(e\)-free if \(\epsilon \neq \beta^d\) for any \(\beta \in \mathbb{F}_{q^m}\) and for any divisor \(d\) of \(e\) other than \(1\).
An element \(\epsilon \in \mathbb{F}^*_{q^m}\) is primitive if and only if it is \((q^m-1)\)-free. The characteristic function for \(e\)-free elements of \(\mathbb{F}^*_{q^m}\) is defined as follows:
\[\label{cfe} \rho_e: \mathbb{F}^*_{q^m} \to \{ 0,1\}; \epsilon \mapsto \theta(e) \sum\limits_{d|e}\frac{\mu(d)}{\phi(d)}\sum\limits_{(d)}\chi_{d}(\epsilon), \tag{1}\] where \(\theta(e):=\frac{\phi(e)}{e}\) and \(\phi, \mu\) denote the Euler’s totient function and Mobius function respectively.
The additive group of \(\mathbb{F}_{q^m}\) is a \(\mathbb{F}_{q}[x]\)-module under the rule \(f \circ\alpha=\sum\limits_{i=0}^{n} a_i\alpha^{q^i}\); for \(\alpha\in \mathbb{F}_{q^m}\) and \(f\,= \sum\limits_{i=0}^n a_ix^i\thinspace \in \mathbb{F}_{q}[x]\). For \(\alpha \in \mathbb{F}_{q^m}\), the \(\mathbb{F}_q\)-order of \(\alpha\) is the monic \(\mathbb{F}_q\)-divisor \(g\) of \(x^m-1\) of minimal degree such that \(g \circ\alpha=0\).
Similarly, for \(\psi \in \widehat{\mathbb{F}_{q^m}}\), the \(\mathbb{F}_q\)-order of \(\psi\) is the monic divisor \(g\) of \(x^m-1\) of minimal degree such that \(\psi \circ g\) is the trivial character, where \((\psi \circ g) (\beta)=\psi(g\circ \beta)\).
Definition 2.3. (\(g\)-free element) Let \(g\) divides \(x^m-1\) and \(\epsilon \in \mathbb{F}_{q^m}\). If \(\epsilon \neq h \circ \gamma\) for any \(\gamma \in \mathbb{F}_{q^m}\) and for any divisor \(h\) of \(g\) other than \(1\), then \(\epsilon\) is called a \(g\)-free element.
An element \(\epsilon \in \mathbb{F}_{q^m}\) is normal if and only if it is \((x^m-1)\)-free. The characteristic function for \(g\)-free elements of \(\mathbb{F}_{q^m}\) is defined as follows: \[\label{cfg} \eta_g: \mathbb{F}_{q^m} \to \{ 0,1\}; \epsilon \mapsto \Theta(g) \sum\limits_{h|g}\frac{\mu^{'}(g)}{\Phi(g)}\sum\limits_{(h)}\psi_{h}(\epsilon), \tag{2}\]
where \(\Theta(g):= \frac{\Phi(g)}{q^{\deg(g)}}\) and \(\Phi(g)=|{(\frac{\mathbb{F}_{q}[x]}{<g>})}^{*}|\) and the function \(\mu^{'}\) is given by:
\(\mu^{'}(g)=\begin{cases} (-1)^s \,\mbox{ if}\, \text{$g$ is a product of $s$ distinct monic irreducible polynomials},\\ 0 \,\mbox{}\, \text{otherwise}. \end{cases}\)
We shall also need characteristic function for prescribed values of trace. For each \(a \in \mathbb{F}_q\) the characteristic function for the elements \(\epsilon \in \mathbb{F}_{q^m}\) such that \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\), is defined as follows:
\[\label{cft} \tau_a: \mathbb{F}_{q^m} \to \{ 0,1\}; \epsilon \mapsto \frac{1}{q}\sum\limits_{\psi \in \widehat{\mathbb{F}_q}}\psi(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)-a). \tag{3}\]
The following lemmas will be handy while finding sufficient conditions. The first result was given by Weil [18] as described in [7] at (1.1) to (1.3).
Lemma 2.4. Let \(f(x)\in \mathbb{F}_{q^m}(x)\) be a rational function. Write \(f(x)=\prod_{j=1}^k {f_j(x)}^{r_j}\), where \(f_j(x) \in \mathbb{F}_{q^m}[x]\) are irreducible polynomials and \(r_j\) are nonzero integers. Let \(\chi\) be a nontrivial multiplicative character of \(\mathbb{F}_{q^m}\) of squarefree order \(d\) (a divisor of \(q^m-1\)). Suppose that \(f(x)\) is not of the form \(cg(x)^d\) for any rational function \(g(x)\in \mathbb{F}_{q^m}(x)\) and \(c \in \mathbb{F}^*_{q^m}\). Then we have,
\[\bigg|\sum\limits_{\alpha\in \mathbb{F}_{q^m},f(\alpha)\neq 0, \infty}\chi(f(\alpha))\bigg| \leq \bigg( \sum\limits_{j=1}^k \deg(f_j)-1\bigg)q^{\frac{m}{2}}.\]
Lemma 2.5. [4] Let \(\chi\) and \(\psi\) be two nontrivial multiplicative and additive characters of the field \(\mathbb{F}_{q^m}\), respectively. Let \(f,g\) be rational functions in \(\mathbb{F}_{q^m} (x)\), where \(f\neq \beta h^r\) and \(g \neq h^p-h+\beta\), for any \(h\in \mathbb{F}_{q^m}(x)\) and any \(\beta\in \mathbb{F}_{q^m}\), and \(r\) is the order of \(\chi\). Then \[\left | \sum\limits_{\alpha\in\mathbb{F}_{q^m} \setminus S}\chi(f(\alpha))\psi(g(\alpha))\right|\leq [\deg(g_\infty)\, +\, l_0 \,+\, l_1\,-\,l_2\,-\,2 ]q^{m/2},\] where \(S\) denotes the set of all poles of \(f\) and \(g\), \(g_\infty\) denotes the pole divisor of \(g\), \(l_0\) denotes the number of distinct zeroes and finite poles of \(f\) in the algebraic closure \(\mathbb{F}\) of \(\mathbb{F}\), \(l_1\) denotes the number of distinct poles of \(g\) (including infinite pole) and \(l_2\) denotes the number of finite poles of \(f\), that are also zeroes or poles of \(g\).
Using these, we can now proceed to find all the pairs \((q,m)\), for which there exists a rational function \(f \in \mathcal{R}_{q^m}^n\) of degree sum \(n\) such that \((\epsilon, f(\epsilon))\) is a primitive normal pair with \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\) and \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\) for some prescribed values \(a,b \in \mathbb{F}^*_q\). Let \(\mathcal{A}_p^n\) denotes the set of all such pairs \((q,m)\), where \(q\) is power some prime \(p\). If \((q,m) \notin \mathcal{A}_p^n\), we refer to such pair as an exceptional pair.
Let \(e_1\), \(e_2\) be two divisors of \(q^m-1\) and \(g_1\), \(g_2\) be two divisors of \(x^m-1\). We denote the number of \(\epsilon \in \mathbb{F}_{q^m}\) such that \(\epsilon\) is both \(e_1\)-free and \(g_1\)-free and \(f(\epsilon)\) is both \(e_2\)-free and \(g_2\)-free with \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\) and \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\), where \(a,b\in \mathbb{F}^*_q\) and \(f(x) \in \mathcal{R}_{q^m}^n\) by \(N_{f,a,b}(e_1,e_2,g_1,g_2)\). Since the trace of an element is nonzero if and only if it is \((x-1)-\)free, therefore to show that an element is \(g_i-\)free, for \(i=1,2\), it is enough to prove that the element is \(L_{g_i}-\)free, where \(L_{g_i}\) is the largest divisor of \(g_i\) such that \((x-1)\nmid L_{g_i}\). Let \(S\) be the set containing all zeros and poles of \(f(x)\) along with \(0\). Now, using the characteristic functions (1), (2), and (3), we get,
\[\begin{aligned} N_{f,a,b}(e_1,e_2,L_{g_1},L_{g_2}) =& \sum\limits_{\epsilon\in \mathbb{F}_{q^m}\mathbb{S}}\rho_{e_1}(\epsilon)\rho_{e_2}(f(\epsilon))\eta_{L_{g_1}}(\epsilon)\eta_{L_{g_2}}(f(\epsilon)) \tau_a(\epsilon)\tau_b(f(\epsilon))\\ =& \frac{\theta(e_1)\theta(e_2)\Theta(L_{g_1})\Theta(L_{g_2})}{q^2} \sum\limits_{\substack{d_1|e_1,d_2|e_2 \\ t_1|L_{g_1},t_2|L_{g_2}}}\frac{\mu(d_1)\mu(d_2)\mu'(t_1)\mu'(t_2)}{\phi(d_1)\phi(d_2)\Phi(t_1)\Phi(t_2)} \\ &\times \sum\limits_{\substack{\chi_{d_1},\chi_{d_2} \\ \psi_{t_1},\psi_{t_2}}}\chi_{f,a,b}(d_1,d_2,t_1,t_2), \end{aligned}\]
where,
\[\begin{aligned} \chi_{f,a,b}(d_1,d_2,t_1,t_2) =& \sum\limits_{\epsilon\in \mathbb{F}_{q^m}\mathbb{S}}\chi_{d_1}(\epsilon)\chi_{d_2}(f(\epsilon))\psi_{t_1}(\epsilon)\psi_{t_2}(f(\epsilon))\sum\limits_{\psi \in \widehat{\mathbb{F}_q}}\psi(T(\epsilon)-a)\\ &\times \sum\limits_{\tilde{\psi} \in \widehat{\mathbb{F}_q}}\tilde{\psi}(T(f(\epsilon))-b),\\ =& \sum\limits_{\epsilon\in \mathbb{F}_{q^m}\mathbb{S}}\chi_{d_1}(\epsilon)\chi_{d_2}(f(\epsilon))\psi_{t_1}(\epsilon)\psi_{t_2}(f(\epsilon))\\ & \times \sum\limits_{u_1 \in \mathbb{F}_q}\bar{\psi}(T(u_1\epsilon)-u_1a) \sum\limits_{u_2 \in \mathbb{F}_q}\bar{\psi}(T(u_2f(\epsilon))-u_2b). \end{aligned}\]
Here we have considered \(u_1, u_2 \in \mathbb{F}_q\), such that \(\psi(\beta)=\bar{\psi}(u_1\beta)\) and \(\tilde{\psi}(\beta)=\bar{\psi}(u_2\beta)\) for all \(\beta \in \mathbb{F}_{q^m}\). Let us denote \(\bar{\psi} (T(\epsilon)) = \psi'(\epsilon), \forall \epsilon \in \mathbb{F}_{q^m}\), which is an additive character of \(\mathbb{F}_{q^m}\). Then the above equation takes the form,
\[\begin{aligned} \chi_{f,a,b}(d_1,d_2,t_1,t_2)=& \sum\limits_{u_1,u_2 \in \mathbb{F}_q}\bar{\psi}(-u_1a-u_2b)\\&\times\sum\limits_{\epsilon\in \mathbb{F}_{q^m}\mathbb{S}} \chi_{d_1}(\epsilon)\chi_{d_2}(f(\epsilon))\psi_{t_1}(\epsilon)\psi_{t_2}(f(\epsilon))\psi'(u_1\epsilon+u_2\epsilon). \end{aligned}\]
There exist, \(v_1,v_2 \in \mathbb{F}_{q^m}\), such that \(\psi_{t_1}(\epsilon)=\psi'(v_1\epsilon)\) and \(\psi_{t_2}(\epsilon)=\psi'(v_2f(\epsilon))\). Moreover we consider \(\chi_{d_1}=\chi^{m_1}_{q^m-1}\) and \(\chi_{d_2}=\chi^{m_2}_{q^m-1}\) for some multiplicative character \(\chi_{q^m-1}\) of order \(q^m-1\), where \(m_1 \in \{ 0,1, \dots ,q^m-2 \}\) and \(m_2 = \frac{q^m-1}{d_2}\). Then we get,
\[\begin{aligned} \chi_{f,a,b}(d_1,d_2,t_1,t_2) =& \sum\limits_{u_1,u_2 \in \mathbb{F}_q}\bar{\psi}(-u_1a-u_2b)\\&\times\sum\limits_{\epsilon\in \mathbb{F}_{q^m}\mathbb{S}}\chi_{q^m-1}(\epsilon^{m_1}f(\epsilon)^{m_2})\psi'(v_1\epsilon+v_2f(\epsilon)+u_1\epsilon+u_2f(\epsilon))\\ =& \sum\limits_{u_1,u_2 \in \mathbb{F}_q}\bar{\psi}(-u_1a-u_2b)\sum\limits_{\epsilon\in \mathbb{F}_{q^m}\mathbb{S}}{\sum} \chi_{q^m-1}(F_1(\epsilon))\psi'(F_2(\epsilon)), \end{aligned}\] where, \[\begin{aligned} F_1(x)=& x^{m_1}f(x)^{m_2},\\ F_2(x)=& (u_1+v_1)x+(u_2+v_2)f(x). \end{aligned}\]
Case I. \(F_2(x) \neq r(x)^{p}-r(x) + \beta\) for any \(r(x)\in \mathbb{F}_{q^m}(x)\), \(\beta \in \mathbb{F}_{q^m}\) and \(F_1(x)=x^{m_1}{f(x)}^{m_2} \neq \beta {(h(x))}^{q^m-1}\) for any \(h(x)\in \mathbb{F}_{q^m}(x)\).
In this case, we can use Lemma 2.5 to derive the following :
\[\big|\chi_{f,a,b}(d_1,d_2,t_1,t_2) \big| \leq q^2(2n+1)q^{m/2}+q^2(|S|-1).\] Case II. \(F_2(x)=h(x)^{p}-h(x)+ \beta\) for some \(h(x) \in \mathbb{F}_{q^m}(x) , \beta \in \mathbb{F}_{q^m}\).
For \(F_2(x)=h(x)^{p}-h(x)+ \beta\) for some \(h(x) \in \mathbb{F}_{q^m}(x), \beta \in \mathbb{F}_{q^m}\); we can show that \(g_1=g_2=1\). For this let us consider \(f(x)=c_1{x^j}\frac{f_1}{f_2}\) and \(h(x)=c_2{x^k}\frac{h_1}{h_2}\), where \(c_1 \in \mathbb{F}^*_{q^m}, c_2 \in \mathbb{F}^*_{q^m}, j,k \in \mathbb{Z}\) and \(f_1, f_2 \in \mathbb{F}_{q^m}[x]\), \(h_1, h_2 \in \mathbb{F}_{q^m}[x]\) are monic polynomials such that \(x\) divides none of them and \(\gcd(f_1,f_2)=1, \gcd(h_1,h_2)=1\). Then \[\begin{aligned} %F_2(x) =& h(x)^{q^m}-h(x)\\ (u_1+v_1)x+(u_2+v_2)f(x)=&h(x)^{p}-h(x) +\beta. \end{aligned}\]
Setting \((u_1+v_1)=k_1\) and \((u_2+v_2)=k_2\) we get, \[\begin{aligned} \label{eq3.1} (k_1xf_2+k_2c_1x^jf_1){h_2^p} =& (c_2^{p}x^{kp}h_1^{p}-c_2x^kh_1h_2^{p-1}+\beta h_2^p){f_2}. \end{aligned} \tag{4}\]
Let \(k_2 \neq 0\), then from (4), we get \(f_2|h_2^{p}\), since \((f_1,f_2)=1\). Also, since \(\gcd(h_2^{p},c_2^{p}x^{kp}h_1^{p}-c_2x^kh_1h_2^{p-1}+\beta h_2^p)=1\), it will imply \(h_2^{p}|f_2\). This further gives us \(f_2=h_2^{p}\), which creates contradiction due to coprimality restrictions i.e., \(\gcd(f_1,f_2)=1, \gcd(h_1,h_2)=1\). Therefore, we must have \(k_2=0\) i.e., \(u_2+v_2=0.\)
So,
\[\begin{aligned}
\label{eq3.2}
k_1x{h_2^p} &= c_2^{p}x^{kp}h_1^{p}-c_2x^kh_1h_2^{p-1}+ \beta
h_2^p,
\end{aligned} \tag{5}\] i.e., \(h_2\)
divides \(c_2^{p}x^{kp}h_1^{p}-c_2x^kh_1h_2^{p-1} +\beta
h_2^p\). This gives a contradiction since
\(\gcd(h_2,c_2^{p}x^{kp}h_1^{p}-c_2x^kh_1h_2^{p-1})=1\).
Therefore, \(k_1=0\) i.e., \(u_1+v_1=0\).
Let us take \(\psi_{t_1}\) to be an additive character of \(\mathbb{F}_q\)-order \(t_1\). Then we will have,
\[\begin{aligned} \psi_{t_1}(t_1 \circ \alpha)=0 \implies \psi'(v_1t_1 \circ \alpha)=0 \implies \psi'(-u_1t_1\circ \alpha)=0, \forall \alpha \in \mathbb{F}_{q^m}. \end{aligned}\]
Since \(u_1 \in \mathbb{F}_q\) and \(\mathbb{F}_q\)-order of \(\psi'\) is \(x-1\), we must have \(x-1|(-u_1t_1)\). Therefore, we get \(u_1=0\). This further implies \(v_1=0\), since \(k_1=0\).
Similarly, from \(k_2=0\), it can be shown that \(u_2=v_2=0\). With this, we arrive at the conclusion that, if \(F_2(x)=h(x)^{p}-h(x)+ \beta\) for some \(h(x) \in \mathbb{F}_{q^m}(x)\) and \(\beta \in \mathbb{F}_{q^m}\), then \(t_1=t_2=1\).
Let \(F_1(x) = x^{m_1}{f(x)}^{m_2} \neq \beta{(h(x))}^{q^m-1}\) for any \(h(x)\in \mathbb{F}_{q^m}(x)\) and \(\beta \in \mathbb{F}_{q^m}\). In this case, we use Lemma 2.4 to prove that \[\big|\chi_{f,a,b}(d_1,d_2,t_1,t_2) \big| \leq q^2nq^{m/2}=nq^{\frac{m}{2}+2}.\]
In case \(F_1(x) = x^{m_1}{f(x)}^{m_2} =
\beta{(h(x))}^{q^m-1}\), for some \(h(x)\in \mathbb{F}_{q^m}(x)\) and \(\beta \in \mathbb{F}_{q^m}\), then \(f(x) \notin \mathcal{R}_{q^m}^n\) due to
violation of the first criterion in the definition.
Case III. \(F_2(x)=h(x)^{p}-h(x)+\beta\) and \(F_1(x)=x^{m_1}{f(x)}^{m_2} =
\beta{(h(x))}^{q^m-1}\), for some \(h(x)\in \mathbb{F}_{q^m}(x)\) and \(\beta \in \mathbb{F}_{q^m}\).
When \(F_1(x)=x^{m_1}{f(x)}^{m_2}=\beta{(h(x))}^{q^m-1}\) for some \(h(x)\in \mathbb{F}_{q^m}(x)\), we can prove that \(d_1=d_2=1\). One can refer to [16] for the proof. So here we have \[(d_1,d_2,t_1,t_2)=(1,1,1,1).\]
Summarizing all the above cases, we observe that \[\big|\chi_{f,a,b}(d_1,d_2,h_1,h_2)\big| \leq (2n+1)q^{\frac{m}{2}+2},\] where \((d_1,d_2,h_1,h_2) \neq (1,1,1,1)\). Now, \[\begin{aligned} N_{f,a,b}(e_1,e_2,L_{g_1},L_{g_2}) \geq & \frac{\kappa}{q^2} \frac{}{} [(q^m-(2n+1)q^{\frac{m}{2}+2})(W(e_1)W(e_2)W(L_{g_1})W(L_{g_2})-1)],\\ \geq & \kappa q^{\frac{m}{2}}[q^{\frac{m}{2}-2}-(2n+1)W(e_1)W(e_2)W(L_{g_1})W(L_{g_2})], \end{aligned}\] where \(\kappa = \theta(e_1)\theta(e_2)\Theta(L_{g_1})\Theta(L_{g_2}).\)
To show that there exists an element \(\epsilon \in \mathbb{F}_{q^m}\) such that for preassigned \(a\), \(b \in \mathbb{F}^*_q\), and a rational function \(f(x) \in \mathcal{R}_{q^m}^n\), \((\epsilon,f(\epsilon))\) is a primitive normal pair with \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\) and \(T_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\), it is enough to show that \(N_{f,a,b}(q^m-1,q^m-1,x^m-1,x^m-1)>0\), which is evident if \[\label{suff} q^{\frac{m}{2}-2}>(2n+1)W(q^m-1)^2W(x^m-1)^2. \tag{16}\]
We will use the following two lemmas to deduce a sieving inequality that will help to improve the sufficient condition (6).
Lemma 4.1. Let \(d\) be a divisor of \(q^m-1\) and \(p_1, p_2,\dots , p_r\) are the remaining distinct primes dividing \(q^m-1\). Furthermore, let \(g\) be a divisor of \(x^m-1\) and \(g_1, g_2, \dots , g_s\) are the remaining distinct irreducible polynomials dividing \(x^m-1\). Then \[\begin{aligned} N_{f,a,b}(q^m-1,q^m-1,x^m-1,x^m-1)\geq \sum\limits_{i=1}^r N_{f,a,b}(p_id,d,g,g)+ \sum\limits_{i=1}^r N_{f,a,b}( d,p_i d, g,g) \\ + \sum\limits_{i=1}^s N_{f,a,b}(d, d,g_i g,g)+ \sum\limits_{i=1}^s N_{f,a,b}(d, d,g,g_ig)-(2r+2s-1)N_{f,a,b}(d, d, g, g). \end{aligned}\]
Lemma 4.2. Let \(d,m,q \in \mathbb{N}, g \in \mathbb{F}_q[x]\) be such that \(q\) is a prime power, \(m \geq 5, d|q^m-1,\) and \(g|x^m-1\). Let \(e\) be a prime number that divides \(q^m-1\) but not \(d\) and \(h\) be any irreducible polynomial dividing \(x^m-1\) but not \(g\). Then we have the following bounds: \[\begin{aligned} \big| N_{f,a,b}(ed,d,g,g)-\theta(e)N_{f,a,b}(d,d,g,g)\big| \leq & (2n+1)\theta(e)\theta(d)^2\Theta(g)^2W(d)^2W(g)^2q^{\frac{m}{2}},\\ \big| N_{f,a,b}(d,ed,g,g)-\theta(e)N_{f,a,b}(d,d,g,g)\big| \leq & (2n+1)\theta(e)\theta(d)^2\Theta(g)^2W(d)^2W(g)^2q^{\frac{m}{2}},\\ \big| N_{f,a,b}(d,d,hg,g)-\Theta(h)N_{f,a,b}(d,d,g,g)\big| \leq & (2n+1)\Theta(h)\theta(d)^2\Theta(g)^2W(d)^2W(g)^2q^{\frac{m}{2}},\\ \big| N_{f,a,b}(d,d,g,hg)-\Theta(h)N_{f,a,b}(d,d,g,g)\big| \leq & (2n+1)\Theta(h)\theta(d)^2\Theta(g)^2W(d)^2W(g)^2q^{\frac{m}{2}}. \end{aligned}\]
Using Lemma 4.1 and Lemma 4.2, we deduce the following:
Theorem 4.3. Let \(d,m,q \in \mathbb{N}, g \in \mathbb{F}_q[x]\) be such that \(q\) is a prime power, \(m\geq 5, d|q^m-1\) and \(g|x^m-1\). Let \(d\) be a divisor of \(q^m-1\) and \(p_1,p_2,\dots, p_r\) be the remaining distinct primes dividing \(q^m-1\). Furthermore, let \(g\) be a divisor of \(x^m-1\) such that \(g_1,g_2,\dots,g_s\) are the remaining distinct irreducible factors of \(x^m-1\). Define: \[l := 1 – 2 \sum\limits_{i=1}^r \frac{1}{p_i} – \sum\limits_{i=1}^s \frac{1}{q^{{\mathrm deg}(g_i)}},\qquad l>0,\] and \[L := \frac{2(r+s)-1}{l}+2.\]
Then \(N_{f,a,b}(q^m-1,q^m-1,x^m-1,x^m-1)>0\) if \[\begin{aligned} \label{primesieve} q^{\frac{m}{2}-2}>(2n+1)W(d)^2W(g)^2L. \end{aligned} \tag{7}\]
Proof. For our convenience, let us denote \(N_{f,a,b}(q^m-1,q^m-1,x^m-1,x^m-1)\) by
\(N\). Then from Lemma 4.1 and 4.2, we have
\[\begin{aligned}
N \geq &\sum\limits_{i=1}^r\{N_{f,a,b}(p_id,d,g,g)-
\theta(p_i)N_{f,a,b}(d,d,g,g)\}\\& +
\sum\limits_{i=1}^r \{N_{f,a,b}( d,p_i d, g,g)
-\theta(p_i)N_{f,a,b}(d,d,g,g)\}\\
&+ \sum\limits_{i=1}^s \{N_{f,a,b}(d, d,g_i g,g)-
\Theta(g_i)N_{f,a,b}(d,d,g,g)\}
\\&+\sum\limits_{i=1}^s \{N_{f,a,b}(d,
d,g,g_ig)-\Theta(g_i)N_{f,a,b}(d,d,g,g)\}\\
&+ \left\{ 2\sum\limits_{i=1}^r\theta
(p_i)+2\sum\limits_{i=1}^s
\Theta (g_i)\right\}N_{f,a,b}(d, d, g, g)
-(2r+2s-1)N_{f,a,b}(d, d, g, g)\\
=& \sum\limits_{i=1}^r\{N_{f,a,b}(p_id,d,g,g)-
\theta(p_i)N_{f,a,b}(d,d,g,g)\}
\\&+ \sum\limits_{i=1}^r\{N_{f,a,b}( d,p_i d, g,g)
-\theta(p_i)N_{f,a,b}(d,d,g,g)\}\\
&+ \sum\limits_{i=1}^s \{N_{f,a,b}(d, d,g_i g,g)-
\Theta(g_i)N_{f,a,b}(d,d,g,g)\}
\\&+\sum\limits_{i=1}^s \{N_{f,a,b}(d,
d,g,g_ig)-\Theta(g_i)N_{f,a,b}(d,d,g,g)\} + lN_{f,a,b}(d,d,g,g)\\
\geq& \theta(d)^2\Theta(g)^2l[\{
-(2n+1)W(d)^2W(g)^2q^{\frac{m}{2}}L\}+\{q^{m-2}-(n+1)+(2n+1)q^{\frac{m}{2}}\}].
\end{aligned}\]
Therefore, when \(l>0\), then \(N_{f,a,b}(q^m-1,q^m-1,x^m-1,x^m-1)>0\)
if \[\begin{aligned}
q^{\frac{m}{2}-2}>(2n+1)W(d)^2W(g)^2L.
\end{aligned} \tag{8}\] ◻
In this section, we particularly study the case when finite fields are of characteristic \(5\) and rational functions are of degree sum \(4\) and try to determine pairs \((q,m) \in \mathcal{A}_5^4\). For large calculations, we use sagemath [15].
The following lemma will be used extensively throughout the process.
Theorem 5.1. ([16], Lemma 2.4).Let \(\nu>0\) be a real number and \(t\) be a positive integer. Then \(W(t)<Dt^{1/ \nu}\), where \(D=\frac{2^r}{(p_1p_2\dots p_r)^{\frac{1}{\nu}}}\) and \(p_1,p_2,\dots,p_r\) are primes \(\leq 2^\nu\) that divide \(t\).
Remark 5.2. For large values of \(q\) and \(m\), calculation of \(W(q^m-1)\) is very time consuming. So, using the above lemma, we find an upper bound for the term \(W(q^m-1)\) as follows: \[W(q^m-1) \leq D(q^m-1)^{\frac{1}{\nu}}<D(q^m)^{\frac{1}{\nu}}.\]
Also, since \(x^m-1\) can have atmost \(m\) linear factors in \(\mathbb{F}_q[x]\). So, \[W(x^m-1)\leq 2^m.\]
Using these two observations, we rewrite our sufficient condition (6) as: \[\label{D} q^{\frac{m}{2}-2}>(2n+1)D^2q^{\frac{2m}{\nu}}2^{2m}. \tag{9}\]
In our case, \(n=4,m\geq 5, q=5^k\) for some positive integer \(k\). We choose \(\nu=21.57\). Then we get \(D\leq 1.52 \times10^{4906}\). For these values, (9) holds for \(k\geq 385897\) and \(m\geq 5\).
| \(\nu\) | \(k\) | \(m_k\) |
|---|---|---|
| 13.7 | 1379,1380, …,385896 | 6 |
| 11.3 | 212, 213, …, 1378 | 7 |
| 10.1 | 84, 85, …, 211 | 8 |
| 9.52 | 48,49, …, 83 | 9 |
| 9.2 | 33,34, …,47 | 10 |
| 8.9 | 26,27,…, 32 | 11 |
| 8.7 | 21,22, …, 25 | 12 |
| 8.6 | 18,19,20 | 13 |
| 8.5 | 16,17 | 14 |
| 8.5 | 14,15 | 15 |
| 8.4 | 13 | 16 |
| 8.4 | 12 | 17 |
| 8.5 | 11 | 18 |
| 8.4 | 10 | 19 |
| 8.4 | 9 | 21 |
| 8.5 | 8 | 24 |
| 8.4 | 7 | 28 |
| 8.5 | 6 | 36 |
| 8.8 | 5 | 52 |
| 9.4 | 4 | 99 |
| 11.3 | 3 | 520 |
Now, for \(3\leq k\leq 385897\), we present a Table 1 that shows suitable values of \(\nu\) and an integer \(m_k\) corresponding to each value of \(k\) such that for all \(m \geq m_k\) and that particular \(\nu\), (9) holds.
Analyzing the values of \(m_k\) and corresponding range of \(k\) from Table 1, we observe that (9) holds for all \(k\geq 48\) and \(m\geq 9\). Based on this, we claim the following:
Lemma 5.3. For \(k\geq 3\), and \(m\geq 9\), \((q,m)\in \mathcal{A}_5^4\).
Proof. Since for \(3\leq k\leq 47\), (9) holds for all \(m\geq m_k\), so for \(9\leq k<m_k\), we first check the following inequality, \[\label{D1} q^{\frac{m}{2}-2}>(2n+1)D^2q^{\frac{2m}{\nu}}(W(x^m-1))^2. \tag{10}\]
This inequality reduces the number of possible exceptional pairs to \(220\). For all these possibilities, we can find suitable values of \(d\) and \(g\) ( as defined in Theorem 4.3), so that (7) holds. ◻
We study the pairs for \(k=1\) and \(k=2\) separately. Let us consider \(m=m'5^j\) for \(j\geq0\) such that \(\gcd(5,m')=1\). Then, we can divide our discussion into the following two cases:
(i) \(m'| q^2-1\),
(ii) \(m' \nmid q^2-1\).
Case I. First, let us consider the pairs \((q,m)\), where \(m'| q^2-1\) and \(k=1\).
Then possible values of \(m'\) are \(1\), \(2\), \(3\), \(4\), \(6\), \(8\), \(12\), \(24\). Since \(W(x^m-1)=W(x^ {m'}-1)\), we can rewrite (9) as \[\label{m'nk1} q^{\frac{m'5^j}{2}-2}>(2n+1)D^2q^{\frac{2m'5^j}{\nu}}2^{2m'}. \tag{11}\]
Then (11) holds when,
(i) \(m'=1\) and \(j\geq 4\),
(ii) \(m'=2\) ,\(3\), \(4\), \(6\) and \(j\geq 3\),
(iii) \(m'=8\) ,\(12\), \(24\) and \(j\geq 2\).
Therefore \((5,m)\in \mathcal{A}_5^4\) unless \(m\) \(=\) \(10\), \(12\), \(15\), \(20\), \(24\), \(25\), \(30\), \(40\), \(50\), \(60\), \(75\), \(100\), \(120\), \(125\), \(150\). For these pairs, we directly check sufficient condition (6) and reduce the list of possible exceptional values of \(m\) to \(10\), \(12\), \(15\), \(20\), \(24\), \(30\). Again for these values of \(m\), we try to find suitable \(d\) and \(g\) ( as defined in Theorem 4.3) for which (7) is satisfied ( see Table 2). But for the pairs \((5,10)\), \((5,12)\), \((5,20)\), \((5,24)\), we could not find such \(d\) and \(g\). So in this case, these are the possible exceptional pairs.
Now let \(k=2\) and \(m' \nmid q^2-1\).
Then possible values of \(m'\) are \(1\), \(2\), \(3\), \(4\), \(6\), \(8\), \(12\), \(13\), \(16\), \(24\), \(26\), \(39\), \(48\), \(52\), \(76\), \(104\), \(156\), \(208\), \(312\), \(624\). We find out that (11) holds when,
(i) \(m'=1\) ,\(2\), \(3\) and \(j\geq 3\),
(ii) \(m'=4\), \(6\), \(8\), \(12\), \(13\), \(16\) and \(j\geq 2\),
(iii) \(m'=24\), \(26\), \(39\), \(48\), \(52\), \(76\), \(104\), \(156\), \(208\), \(312\), \(624\) and \(j\geq 1\).
Therefore \((25,m) \in \mathcal{A}_5^4\) unless \(m=10\), \(12\), \(13\), \(15\), \(16\), \(20\), \(24\), \(25\), \(26\), \(30\), \(39\), \(40\), \(48\), \(50\), \(52\), \(60\), \(65\), \(75\), \(76\), \(80\), \(104\), \(156\), \(208\), \(312\), \(624\). We reduce these exceptions by removing the pairs \((25,m)\) that satisfy (6). This leaves us with possible exceptional values of \(m=10,12,13,15,16,24,48.\) For these pairs, we try to find \(d\) and \(g\)(as defined in Theorem (4.3)) such that (7) holds( see Table 2). Since we fail to find such \(d\) and \(g\) for \(m=10,12,24\); so \((25,10),(25,12),(25,24)\) adds up to the list of exceptional pairs.
| SL No. | \((q,m)\) | \(d\) | \(r\) | \(g\) | \(s\) | \(l\) | \(L\) |
|---|---|---|---|---|---|---|---|
| 1 | (5,13) | 2 | 1 | \((x+4)\) | 3 | 0.990 | 9.07 |
| 2 | (5,15) | 2 | 5 | \((x+4)\) | 1 | 0.633 | 19.37 |
| 3 | (5,21) | 2 | 4 | \((x+4)\) | 4 | 0.850 | 19.65 |
| 4 | (5,22) | 6 | 4 | \((x+1)\) | 5 | 0.480 | 37.04 |
| 5 | (5,30) | 462 | 7 | \((x+1)\) | 3 | 0.298 | 65.71 |
| 6 | (5,80) | 66 | 12 | \((x+1)(x+2)(x+3)(x+4)\) | 4 | 0.466 | 68.50 |
| 7 | \((5^2,11)\) | 6 | 4 | \((x+4)\) | 2 | 0.883 | 14.46 |
| 8 | \((5^2,13)\) | 6 | 3 | \((x+4)\) | 6 | 0.980 | 19.34 |
| 9 | \((5^2,14)\) | 6 | 5 | \((x+1)\) | 5 | 0.692 | 29.45 |
| 10 | \((5^2,15)\) | 6 | 9 | \((x+4)\) | 2 | 0.231 | 93.03 |
| 11 | \((5^2,18)\) | 42 | 8 | \((x+1)\) | 9 | 0.216 | 155.00 |
| 12 | \((5^2,21)\) | 6 | 10 | \((x+4)\) | 8 | 0.348 | 102.67 |
| 13 | \((5^2,36)\) | 546 | 12 | \((x+1)(x+2)(x+3)(x+4)\) | 16 | 0.095 | 579.62 |
| 14 | \((5^2,48)\) | 9282 | 10 | \(g'\) | 22 | 0.066 | 716.32 |
Here \(g'=(x + 1)(x + 2)(x + 3)(x + 4)(x + \beta)(x + \beta + 1) (x + \beta + 2)(x + \beta + 3)(x + \beta + 4)(x + 2\beta)(x + 2\beta + 1)(x + 2\beta + 2)(x + 2\beta + 3)(x + 2\beta + 4),\) with \(\beta\) as an generating element of the field.
Before moving to the second case, let us recall the following lemmas.
Lemma 5.4([11], Lemma 6.1). Suppose \(q=5^k, m'>4\) and \(\bar{m}=\gcd(q-1, m')\). If \(M'\) denotes the number of distinct irreducible factors of \(x^{m'} -1\) over \(\mathbb{F}_q\), \(e\) is the order of \(q(\mod {m'})\) and \(\delta(q,m)= \frac{M'}{e}\); Then, the following hold.
\(\delta(q,m)\leq \frac{1}{2}\), for \(m=2\bar{m}\),
\(\delta(q,m)\leq \frac{3}{8}\), for \(m= 4\bar{m}\),
\(\delta(q,m)\leq \frac{13}{36}\), for \(m= 6\bar{m}\),
\(\delta(q,m)\leq \frac{1}{3}\), otherwise.
Lemma 5.5 ([16], Lemma 5.3). Assume that \(q=p^k,k \in \mathbb{N}\) and \(m=m'p^j, j\geq 0\) is a positive integer such that \(m'\nmid q-1\) and \(\gcd(m',p)=1\). Let \(e (>2)\) denote the order of \(q \mod m'\). Then \(d=q^m-1\) and \(g\) as the product of the irreducible factors of \(x^{m'}-1\) of degree less than \(e\). Then, in the notation of Theorem 4.3, we have \(L\leq 2m'\).
Since \(m'\nmid q^2-1\) implies \(m'\nmid q-1\) and \(e>2\), we can transform (9) using the above two lemmas as follows:
\[q^{\frac{m'5^j}{2}-2}>9D^2q^{\frac{2m'5^j}{\nu}}2^{2m'\delta(q,m')}2m',\] which is also true if \[\label{del} q^{\frac{m}{2}-2}>18D^2q^{\frac{2m}{\nu}}2^{2m\delta(q,m')}m. \tag{12}\]
Now, we can proceed to the second case.
Case II. \(m' \nmid q^2-1.\) Let \(k=1\). Then possible values of \(m'\) are \(5\), \(7\), \(9\), \(10\), \(11\), \(13\), \(14\), \(15\), \(16\), \(17\), \(18\), \(19\), \(20\), \(21\), \(22\), \(23\). Out of these only \(16\) is there, such that \(16=4.\bar{m}\). So for others, we can rewrite (13) as \[ 5^{\frac{m}{2}-2}>18D^2m5^{\frac{2m}{\nu}}2^{\frac{2m}{3}}. \tag{13}\]
For \(\nu =11.3\), (13) holds for \(m\geq 1566\). So we list out the possible exceptional values of \(m\) in this case to be \(m= 9\), \(10\), \(11\), \(13\), \(14\), \(15\), \(17\), \(18\), \(20\), \(21\), \(22\), \(23\), \(25\), \(35\), \(45\), \(50\), \(55\), \(65\), \(70\), \(75\), \(85\), \(90\), \(95\), \(100\), \(105\), \(110\), \(115\), \(125\), \(175\), \(225\), \(250\), \(275\), \(325\), \(350\), \(375\), \(425\), \(450\), \(475\), \(500\), \(525\), \(550\), \(575\), \(625\), \(875\), \(1125\), \(1250\), \(1375\). Out of them \(m=9\), \(10\), \(11\), \(13\), \(14\), \(15\), \(18\), \(20\), \(21\), \(22\) don’t satisfy (6). For these values, we try to calculate \(d\) and \(g\) so that condition (7) is satisfied ( see Table 2). For \(m=9\), \(10\), \(11\), \(14\), \(18\) and \(20\), we fail to do so. Therefore, they also belong to the list of exceptional values of \(m.\)
The case \(m'=16\) is handled separately. For this (11) takes the form \[\label{del2} 5^{\frac{m}{2}-2}>18D^2m5^{\frac{2m}{\nu}}2^{\frac{3m}{4}}. \tag{14}\]
For \(\nu =11.3\), (14) holds for \(m\geq 342\). This leaves us with \(m=16\), \(80\). But for \(m=16\), neither (6) holds, nor can we find any \(d\) and \(g\) to satisfy (7). So \((5,16)\) is also a possible exceptional pair. With this, we exhaust the case for \(k=1.\)
For \(k=2\), there is no \(m'\) such that \(m'=2\bar{m},4\bar{m}\) or \(6\bar{m}\). So here we can rewrite (11) as \[\label{del3} (25)^{\frac{m}{2}-2}>18D^2m{(25)}^{\frac{2m}{\nu}}2^{\frac{2m}{3}}. \tag{15}\]
With \(\nu =11.3\), (15) holds for all \(m\geq 159\).
Therefore, we need to check for \(m \in \{ 9,10,..,158\} \backslash \{12,13,16,24,26,39,48,52,78,104,156\}.\) Out of these, for \(m=9,\) \(10\), \(11\), \(14\), \(15\), \(18\), \(21\) and \(36\), (6) do not hold. So we search for \(d\) and \(g\) so that (7) holds for them. But for \(m=9\), \(10\), we could not find such \(d\) and \(g\), so \((25,9)\), \((25,10)\) are added up to the list of possible exceptional pairs.
In summarizing the discussions in this section, we obtain the proof of Theorem 1.3.
The first author is supported by NFOBC fellowship (NBCFDC Ref. No. 231610154828). The Second author is supported by DST INSPIRE Fellowship, under grant no. DST/INSPIRE Fellowship/2021/IF210206.