Abstract
Let p be an odd prime. For \(b,c\in {\mathbb {Z}}\), we study the Legendre symbol \(\big (\frac{D_p^*(b,c)}{p}\big )\), where \(D_p^*(b,c)\) denotes the determinant of the matrix \([(i^2+bij+cj^2)^{p-3}]_{1\le i,j\le p-1}\). For example, we prove that if \(p\equiv 2\ (\mathrm{{mod}}\ 3)\) then
for some integer \(x\not \equiv 0\ (\mathrm{{mod}}\ p)\). We also show that
if \(p\equiv 7\ (\mathrm{{mod}}\ 8)\).
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1 Introduction
For an \(n\times n\) matrix \([a_{ij}]_{1\le i,j\le n}\) over a commutative ring with identity, we use \(\det |a_{ij}|_{1\le i,j\le n}\) or \(|a_{ij}|_{1\le i,j\le n}\) to denote its determinant.
Let p be an odd prime, and let \(b,c\in {\mathbb {Z}}\). Sun [4] introduced
and
and proved the following results:
and
where \((\frac{\cdot }{p})\) denotes the Legendre symbol. Grinberg, Sun and Zhao [1, Theorem 1.3] determined \((\frac{S_c(b,p)}{p})\) in the case \(p\not \mid bc\), where
For each prime \(p\equiv 5\ (\mathrm{{mod}}\ 6)\), Sun [4] conjectured that
is a quadratic residue modulo p. This was confirmed by Wu et al. [8].
For any odd prime p and a p-adic integer \(x\not \equiv 0\ (\mathrm{{mod}}\ p)\), clearly
Let p be an odd prime, and let \(b,c\in {\mathbb {Z}}\). Sun [5] showed that for any integer n with \((p-1)/2<n<p-1\) we have
Sun [5] also introduced
and proved that for any prime \(p>3\) with \(p\equiv 3\ (\mathrm{{mod}}\ 4)\) we have
By Wu et al. [8], we actually have \(\left( \frac{D_p(1,1)}{p}\right) =\left( \frac{-2}{p}\right) \) if \(p\equiv 2\ (\mathrm{{mod}}\ 3)\). Recently, Luo and Sun [3] have proved that
and that
Their tools include generalized trinomial coefficients and Lucas sequences. Similar to (1.1), Wu and She [7] extended a result of Sun [5] by proving that \(D_p(b,c)\equiv 0\ (\mathrm{{mod}}\ p)\) if \((\frac{c}{p})=-1\).
We first present a basic result which is similar to (1.1) and Sun [5, Theorem 1.2].
Theorem 1.1
Let p be an odd prime, and let \(b,c\in {\mathbb {Z}}\) with \((\frac{c}{p})=-1\). For any integer n in the interval \([1,p-1]\), we have
Proof
For \(j=1,\ldots ,p-1\), let \(\pi _c(j)=\{cj\}_p\), the least nonnegative residue of cj modulo p. By Zolotarev’s Lemma (cf. [9]), the sign of \(\pi _c \in S_{p-1}\) is exactly the Legendre symbol \((\frac{c}{p})\). Observe that
Thus, with the aid of Fermat’s little theorem, we obtain (1.4). \(\square \)
Let p be an odd prime, and let \(b,c\in {\mathbb {Z}}\). In contrast to the notation \(D_p(b,c)\), we introduce
If \((\frac{b^2-4c}{p})=-1\), then \(i^2+bij+cj^2\not \equiv 0\ (\mathrm{{mod}}\ p)\) for all \(i,j=1,\ldots ,p-1\), and hence,
The notations \(D_p(b,c)\) and \(D_p^*(b,c)\) are motivated by Wolstenholme’s congruences (cf. [6])
provided \(p>3\).
Now we state our main results.
Theorem 1.2
Let p be an odd prime. Then
Consequently, when \(p\equiv 2\ (\mathrm{{mod}}\ 3)\) the p-adic integer
is a quadratic residue modulo p.
Theorem 1.3
Let p be an odd prime. Then
Remark 1.1
Note that for any prime \(p\equiv 3\ (\mathrm{{mod}}\ 4)\) we have
Let \(n\in {\mathbb {N}}=\{0,1,2,\ldots \}\) and \(b,c\in {\mathbb {Z}}\). The generalized trinomial coefficients
are given by
We will make use of generalized trinomial coefficients to prove Theorems 1.2 and 1.3 in Sects. 2 and 3, respectively. Note that Theorems 1.2 and 1.3 cannot be deduced from Luo and Sun’s results (1.2) and (1.3), and their proofs are somewhat sophisticated.
2 Proof of Theorem 1.2
Lemma 2.1
([2, Lemma 10]) Let R be a commutative ring with identity, and let \(P(x)=\sum _{i=0}^{n-1}a_ix^i\in R[x]\). Then
Lemma 2.2
(Luo and Sun [3, (3.2)]) For any odd prime p, we have
Lemma 2.3
([3, Lemma 2.1]) Let p be an odd prime, and let \(b,c\in {\mathbb {Z}}\). For \(k\in \{-p+2,\ldots ,p-2\}\), we have
Lemma 2.4
Let p be an odd prime, and let \(b,c\in {\mathbb {Z}}\). For \(k\in \{-p+3,\ldots ,p-3\}\), we have
Proof
For \(n\in {\mathbb {N}}\) and \(k\in {\mathbb {Z}}\), we simply write \({n\brack k}\) for \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _{b,c}\).
Taking derivatives of both sides of the following identity
we get
Comparing the coefficients of \(x^{k-1}\) on both sides of (2.4), we obtain
Taking derivatives of both sides of (2.4), we get
Comparing the coefficients of \(x^{k-2}\) on both sides of (2.6), we deduce that
For \(n\in {\mathbb {Z}}^+=\{1,2,3,\ldots \}\) and \(k\in {\mathbb {Z}}\), we have the recurrence
by Luo and Sun [3, (2.3)]. With the aid of this, we have
and hence,
Combining (2.5), (2.7) and (2.8), we get
Hence, we have
that is,
This concludes the proof. \(\square \)
Proof of Theorem 1.2
Let \(b,c\in {\mathbb {Z}}\). By Luo and Sun [3, (2.2)], we have
Thus,
Let \(k\in \{-p+3,\ldots ,p-3\}\). Taking \(b=c=1\) in Lemma 2.4, we get
Putting \(b=c=1\) in (2.2) and noting (2.10), we obtain
Combining (2.12) with (2.13), we see that
For each \(k\in \{0,\ldots ,p-3\}\), we have
by Luo and Sun [3, (2.14)], and hence,
When \(2\le k\le p-3\), we have
and hence,
In view of (2.11) and (2.14), we obtain
Thus, with the aid of (2.15), we have
Let \(F(x)=(x^2+x+1)^{p-3}\). For \(1\le i, j\le p-1\), we have
and hence,
by Fermat’s little theorem.
Case 1. \(p\equiv 1\ (\mathrm{{mod}}\ 3)\).
Applying Lemma 2.1 with \(P(x)=F(x)\), \(X_i=i\) and \(Y_j=1/j\), and noting the identity (2.1), we get
and hence,
Observe that
Combining (2.17)–(2.19), we obtain
Case 2. \(p\equiv 2\ (\mathrm{{mod}}\ 3)\).
By Lemma 2.1 and the identity (2.1), we have
Combining this with (2.17), we finally obtain
In view of the above, we have completed our proof of Theorem 1.2. \(\square \)
3 Proof of Theorem 1.3
Lemma 3.1
Let \(p>5\) be a prime, and let \(b,c\in {\mathbb {Z}}\). Then
where
Proof
Let \(G(x)=(x^2+bx+c)^{p-3}\). For \(1\le i, j\le p-1\), we have
and hence,
by Fermat’s little theorem. In view of (2.11), and Lemmas 2.1 and 2.2, we see that
Since
for all \(k=2,\ldots ,(p-3)/2\), we have
Combining the above, we immediately obtain the desired identity (3.1). \(\square \)
Proof of Theorem 1.3
Applying Theorem 1.1 with \(c=2\), we see that \((\frac{D_p^*(2,2)}{p})=0\) if \(p\equiv \pm 3\ (\mathrm{{mod}}\ 8)\). Below we assume that \(p\equiv \pm 1\ (\mathrm{{mod}}\ 8)\).
Let \(k\in \{0,\ldots ,p-3\}\). Taking \(b=c=2\) in Lemma 2.4 and (2.2), we get
and
Combining (3.2) and (3.3), we have
and also
if \(2\le k\le p-3\).
For any \(2\le k\le p-3\), define
Then
Define the sequence \((u_n)_{n\ge 0}\) by
By Luo and Sun [3, (4.3)],
for all \(k=0,1,\ldots ,p-1\). Thus,
for all \(k=2,\ldots ,p-3\).
Clearly, (3.5) with \(k=(p-1)/2\) yields that
By Luo and Sun [3, (4.9)], for any \(k\in {\mathbb {N}}\) we have
Case 1. \(p\equiv 1\ (\mathrm{{mod}}\ 8)\).
Write \(p=8q+1\) with \(q\in {\mathbb {N}}\). In view of (3.8) and (3.9), we have
and hence,
Therefore,
Note that
by (3.10). Combining this with Lemma 3.1, (3.12) and (3.11), we obtain
Case 2. \(p\equiv 7\ (\mathrm{{mod}}\ 8)\).
In this case, we write \(p=8q+7\) with \(q\in {\mathbb {N}}\). For \(2\le k\le p-3\), write \(k=4s+r\) with \(s\in {\mathbb {N}}\) and \(r\in \{0,1,2,3\}.\) We will first show that \(W(k)\not \equiv 0\ (\mathrm{{mod}}\ p)\) for any \(k\in \{2,3,\ldots ,p-3\}\).
Subcase 2.1. \(r=0\).
In this subcase, by (3.7), (3.9) and Fermat’s little theorem, we have
Subcase 2.2. \(r=1\).
In this subcase, by (3.7) and (3.9) we have
Subcase 2.3. \(r=2\).
In view of (3.7) and (3.9), we have
Subcase 2.4. \(r=3\).
In this subcase, by (3.7) and (3.9) we have
In view of the discussions for all the four subcases, we do have
and
Therefore,
and hence,
Therefore,
Combining Lemma 3.1, (3.12)–(3.14), we finally obtain
In view of the above, we have finished our proof of Theorem 1.3. \(\square \)
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Acknowledgements
The authors are indebted to the two anonymous referees for their helpful comments. The research is supported by the National Natural Science Foundation of P. R. China (Grant No. 12371004).
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Wang, H., Sun, ZW. On Certain Determinants and Related Legendre Symbols. Bull. Malays. Math. Sci. Soc. 47, 58 (2024). https://doi.org/10.1007/s40840-024-01650-2
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DOI: https://doi.org/10.1007/s40840-024-01650-2