## What is/are Odd Prime?

Odd Prime - Let $p$ be an odd prime and let $J_o(X)$, $J_r(X)$ and $J_e(X)$ denote the three different versions of Thompson subgroups for a $p$-group $X$.^{[1]}In this paper, we introduce the notion of a quasi-powerful p-group for odd primes p.

^{[2]}Let [Formula: see text] be an odd prime and [Formula: see text].

^{[3]}Such an algorithm allows us to greatly extend the numerical investigations about the Euler-Kronecker constants $\mathfrak{G}_q$, $\mathfrak{G}_q^+$ and $M_q=\max_{\chi\ne \chi_0} \vert L^\prime/L(1,\chi)\vert$, where $q$ is an odd prime, $\chi$ runs over the primitive Dirichlet characters $\bmod\ q$, $\chi_0$ is the trivial Dirichlet character $\bmod\ q$ and $L(s,\chi)$ is the Dirichlet $L$-function associated to $\chi$.

^{[4]}Let [Formula: see text] be a positive integer and let [Formula: see text] be an odd prime.

^{[5]}In this paper, we construct some balanced quaternary sequences of odd period p with low autocorrelation from two types of Legendre sequences and its cyclic shift or complement and inverse Gray mapping, where p is odd prime.

^{[6]}Let p be an odd prime.

^{[7]}As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

^{[8]}In 2003, Rodriguez–Villegas found four supercongruences modulo $$p^2$$ (p is an odd prime) for truncated $$_3F_2$$ hypergeometric series related to Calabi–Yau manifolds of dimension $$d=3$$.

^{[9]}org/1998/Math/MathML">

^{[10]}In 1997, Helleseth and Sandberg proved that the differential uniformity of $x^{\frac {p^{n}-1}{2}+2}$ over $\mathbb {F}_{p^{n}}$ , where p is an odd prime, is less than or equal to 4.

^{[11]}In this paper, we study skew cyclic codes over the ring [Formula: see text] where [Formula: see text]; [Formula: see text] is an odd prime.

^{[12]}Let

^{[13]}Let E be an elliptic curve over an imaginary quadratic field K, and p be an odd prime such that the residual representation E [ p ] is reducible.

^{[14]}Let $p$ be an odd prime, and let $m$ be a positive integer satisfying $p^m \equiv 3~(\text{mod }4).

^{[15]}In this paper, we show that $f(k)$ exists if and only if $k$ is an odd prime.

^{[16]}In this paper, we completely classify the finite p-groups with $$\nu _c=p$$ or $$p+1$$ for an odd prime number p.

^{[17]}In this paper, for an odd prime $p$, by extending Li et al.

^{[18]}Particular attention is given to values of the general forms SS(2mp), SS(6mp), SS(60mp) and SS(420mp), where m is any (positive) integer and p is an odd prime.

^{[19]}We focus on the dimension two and study the quality of the simultaneous approximation to two p-adic numbers provided by p-adic MCFs, where p is an odd prime.

^{[20]}We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group.

^{[21]}Let p be an odd prime number and g < 2 be an integer.

^{[22]}org/1998/Math/MathML" id="M1">

^{[23]}We prove that it will suffice to take a finite quotient that is Abelian, dihedral, a subgroup of PSL(n,Fq) (for an odd prime power q), or an Abelian extension of one of these 3 groups.

^{[24]}In this paper, we introduced certain formulas for p-adic valuations of Stirling numbers of the second kind S(n, k) denoted by vp(S(n, k)) for an odd prime p and positive integers k such that n ≥ k.

^{[25]}Let p be an odd prime, and let k be a nonzero nature number.

^{[26]}The main purpose of this article is using the elementary methods and the properties of the quadratic residue modulo an odd prime to study the calculating problem of the fourth power mean of one kind two-term exponential sums and give an interesting calculating formula for it.

^{[27]}Heilbronn sums is of the form H_p(a)=\underset{l=1}{\overset{p-1}{\sum}}e(\dfrac{al^p}{p^2}), where p is an odd prime, and e(x)=\exp(2\pi ix).

^{[28]}Let 𝔽q{\mathbb{F}_{q}} be the finite field of order q, where q is an odd prime power.

^{[29]}For $g=12$, we present lower and upper bounds for this invariant when $q\ge 9$ an odd prime power.

^{[30]}Moreover, we also compute the explicit values of \begin{document}$ \eta_i^{(2N, q)} $\end{document} , \begin{document}$ i = 0,1,\cdots, 2N-1 $\end{document} , if \begin{document}$ q $\end{document} is a power of an odd prime \begin{document}$ p $\end{document}.

^{[31]}We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,\lambda)$ design with $\gcd(k,\lambda)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq A\Gamma L_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$.

^{[32]}

For an odd prime ^{[33]}
Let p be an odd prime number.
^{[34]}
For odd primes $p$, we construct closed locally CAT(0) manifolds with nonzero mod $p$ homology growth outside the middle dimension.
^{[35]}
We consider the calculation problem of one kind hybrid power mean involving the character sums of polynomials and two-term exponential sums modulo $ p $, an odd prime, and use the analytic method and the properties of classical Gauss sums to give some identities and asymptotic formulas for them.
^{[36]}
For an odd prime p and q = pr, this paper deals with LCD codes obtained from cyclic codes of length n over a finite commutative non-chain ring $\mathcal {R}=\mathbb {F}_{q}[u,v]/\langle u^{2}-\alpha u,v^{2}-1, uv-vu\rangle $
where α is a non-zero element in $\mathbb {F}_{q}$.
^{[37]}
We in this paper determines a formula for the number of subgroups, normal and cyclic subgroups of the group G = D 2n × C p = 〈a, b, c|a n = b 2 = c p , b a b = a −1, [a, c] = [b, c] = 1〉, where p is an odd prime number.
^{[38]}
Let $p=2^{e+1}q+1$ be an odd prime number with $2 \nmid q$.
^{[39]}
Let $G = PSL_{2}(q)$, where $q$ is a power of an odd prime.
^{[40]}
We compute their Lefschetz numbers, and show that the Lefschetz numbers of period m are non-zero, for all m ’s, in the case that n is an odd prime.
^{[41]}
We extend the results of Amir and Hong in \cite{AH} for $k=2$ by ruling out or locating all odd prime values $|\ell|<100$ of their Fourier coefficients $a(n)$ when $n$ satisfies some congruences.
^{[42]}
For any positive integer m > 2 and an odd prime p, let $\mathbb {F}_{p^{m}}$
be the finite field with pm elements and let $ \text {Tr}^{m}_{e}$
be the trace function from $\mathbb {F}_{p^{m}}$
onto $\mathbb {F}_{p^{e}}$
for a divisor e of m.
^{[43]}
In the present paper, we prove that, for an odd prime number $p$ and a positive integer $g$ such that $g-1$ is divisible by $p$, there exists a Tango curve of genus $g$ in characteristic $p$.
^{[44]}
Guo $$\begin{aligned}&\sum _{k=0}^{p-1}(-1)^k(2k+1)P_k(2x+1)^4\\ \equiv&p\sum _{k=0}^{(p-1)/2}(-1)^k\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2(x^2+x)^k(2x+1)^{2k}\pmod {p^3},\end{aligned}$$
where p is an odd prime and x is an integer.
^{[45]}
Let p be an odd prime and the integral group ring of the dihedral group of order 2p.
^{[46]}
Sequences are formed on the basis of new generalized cyclotomy modulo equal to the degree of an odd prime.
^{[47]}
We consider the subgroups $ H $ in a symplectic or orthogonal group over a finite field of odd characteristic such that $ O_{r}(H)\neq 1 $ for some odd prime $ r $.
^{[48]}
Let A be an abelian variety defined over a number field k, let p be an odd prime number and let
$F/k$
be a cyclic extension of p-power degree.
^{[49]}
Let p be an odd prime, and k an integer such that k ∣ ( p − 1 ).
^{[50]}

## entanglement assisted quantum

In this work, by investigating the decomposition of the defining set of constacyclic codes, we obtain two types of q-ary entanglement-assisted quantum MDS(EAQMDS) codes with length $$n=\frac{q^2+1}{10\mu }$$ , where m is a positive integer, q is an odd prime power such that $$q=10\mu m+\nu $$ or $$q=10\mu m+10\mu -\nu $$ , and both $$\mu $$ and $$\nu $$ are odd with $$10\mu =\nu ^2+1$$ and $$\nu \ge 3$$.^{[1]}In this paper, we utilize the decomposition of the defining set and

^{[2]}

## primitive roots modulo

In this note, we refine Gauss’s famous theorem on the existence of primitive roots modulo pℓ for every odd prime number p and for every integer and observe the following: For an odd prime number , at least half of the primitive roots modulo p are primitive roots modulo pℓ for every integer.^{[1]}

## Distinct Odd Prime

In this paper, we study a family of the binary sequences derived from Euler quotients modulo pq , where p and q are two distinct odd primes and p divides q − 1.^{[1]}In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers.

^{[2]}In this work, we present the integral trace form [Formula: see text] of a cyclic extension [Formula: see text] with degree [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct odd primes, the conductor of [Formula: see text] is a square free integer, and [Formula: see text] belongs to the ring of algebraic integers [Formula: see text] of [Formula: see text].

^{[3]}In this paper, intrigued by De Koninck and Luca’s work, we further determine all weakly prime-additive numbers [Formula: see text] such that [Formula: see text], where [Formula: see text] are distinct odd prime factors of [Formula: see text].

^{[4]}In this paper, we study quadratic residue (QR) codes of prime length q over the ring R = 𝔽p + u𝔽p + v𝔽p with u2 = u,v2 = v and uv = vu = 0, where p and q are distinct odd prime numbers.

^{[5]}

## Every Odd Prime

We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$.^{[1]}We prove that, for every odd prime number [Formula: see text], there are [Formula: see text] paramedial quasigroups of order [Formula: see text] and [Formula: see text] paramedial quasigroups of order [Formula: see text], up to isomorphism.

^{[2]}In this note, we refine Gauss’s famous theorem on the existence of primitive roots modulo pℓ for every odd prime number p and for every integer and observe the following: For an odd prime number , at least half of the primitive roots modulo p are primitive roots modulo pℓ for every integer.

^{[3]}

## Given Odd Prime

We introduce admissible collections for a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the Quillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of |G|\lvert G\rvert.^{[1]}For a given odd prime number p, in this paper we construct a minimal generating set for the mod-p cohomology of the Steinberg summand of a family of Thom spectra over the classifying space of an elementary p-abelian group.

^{[2]}

## odd prime number

In this paper, we completely classify the finite p-groups with $$\nu _c=p$$ or $$p+1$$ for an odd prime number p.^{[1]}Let p be an odd prime number and g < 2 be an integer.

^{[2]}In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers.

^{[3]}For a given odd prime number p, in this paper we construct a minimal generating set for the mod-p cohomology of the Steinberg summand of a family of Thom spectra over the classifying space of an elementary p-abelian group.

^{[4]}We prove that, for every odd prime number [Formula: see text], there are [Formula: see text] paramedial quasigroups of order [Formula: see text] and [Formula: see text] paramedial quasigroups of order [Formula: see text], up to isomorphism.

^{[5]}Let p be an odd prime number.

^{[6]}We in this paper determines a formula for the number of subgroups, normal and cyclic subgroups of the group G = D 2n × C p = 〈a, b, c|a n = b 2 = c p , b a b = a −1, [a, c] = [b, c] = 1〉, where p is an odd prime number.

^{[7]}Let $p=2^{e+1}q+1$ be an odd prime number with $2 \nmid q$.

^{[8]}In this paper, we explain all non-negative integer solutions for the nonlinear Diophantine equation of type 8x + py = z2 when p is an arbitrary odd prime number and incongruent with 1 modulo 8.

^{[9]}In the present paper, we prove that, for an odd prime number $p$ and a positive integer $g$ such that $g-1$ is divisible by $p$, there exists a Tango curve of genus $g$ in characteristic $p$.

^{[10]}Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree.

^{[11]}We construct an explicit class of affine permutation matrix low-density parity-check (APM-LDPC) codes based on the array parity-check matrix by using two affine maps f (x) = x-1 and g(x) = 2x-1 on Z_m, where m is an odd prime number, with girth 6 and flexible row (column)-weights.

^{[12]}Let p be any odd prime number and let m, k be arbitrary positive integers.

^{[13]}Let $${\mathcal{R}}$$ R be the finite chain ring $${\mathcal{R}}={\mathbb{F}}_{p^{m}}+ u{\mathbb{F}}_{p^{m}}(u^{2} = 0)$$ R = F p m + u F p m ( u 2 = 0 ) , where p is an odd prime number and m is a positive integer.

^{[14]}The first result of this research is the form of the coprime graph of a generalized quaternion group Q _(4 n ) when n = 2^k, n an odd prime number, n an odd composite number, and n an even composite number.

^{[15]}Consider an odd prime number p ≡ 2 ( mod 3 ) p\equiv 2\hspace{0.

^{[16]}We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn) for n = 2kpℓ, where k, ℓ ≥ 0 are integers and p is an odd prime number.

^{[17]}In this paper, we study quadratic residue (QR) codes of prime length q over the ring R = 𝔽p + u𝔽p + v𝔽p with u2 = u,v2 = v and uv = vu = 0, where p and q are distinct odd prime numbers.

^{[18]}For example, the first member (Fermat-1) in this collection of sequences is odd prime number sequence, which has factor–of course–of 1 and has highest number of iteration in the FFM, which is ~half of the number itself.

^{[19]}In this paper, we prove this conjecture for $$S\in \{\mathrm{PSL}_{2}(p^{f}), \mathrm{PSL}_{2}(2^{f}), \mathrm{Sz}(q)\}$$, where $$p>2$$ is an odd prime number such that $$p^{f}>5$$ and $$p^{f}\pm 1\not \mid 2^{k}$$, and $$q=2^{2n+1}\geqslant 8$$.

^{[20]}Let l be a regular odd prime number, k the lth cyclotomic field, k∞ the cyclotomic ℤl-extension of k, K a cyclic extension of k of degree l, and = K · k∞.

^{[21]}Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field which has good reduction at every prime above $p$.

^{[22]}With the obtained sets, the reduced graphs and complementary trees were established some properties that are analyzed in routines developed with Mathematica, providing a visual interpretation of the structures, object of the study, and allowing several tests with different values for odd prime number p.

^{[23]}In this note, we refine Gauss’s famous theorem on the existence of primitive roots modulo pℓ for every odd prime number p and for every integer and observe the following: For an odd prime number , at least half of the primitive roots modulo p are primitive roots modulo pℓ for every integer.

^{[24]}

## odd prime power

In this work, by investigating the decomposition of the defining set of constacyclic codes, we obtain two types of q-ary entanglement-assisted quantum MDS(EAQMDS) codes with length $$n=\frac{q^2+1}{10\mu }$$ , where m is a positive integer, q is an odd prime power such that $$q=10\mu m+\nu $$ or $$q=10\mu m+10\mu -\nu $$ , and both $$\mu $$ and $$\nu $$ are odd with $$10\mu =\nu ^2+1$$ and $$\nu \ge 3$$.^{[1]}Let

^{[2]}We prove that it will suffice to take a finite quotient that is Abelian, dihedral, a subgroup of PSL(n,Fq) (for an odd prime power q), or an Abelian extension of one of these 3 groups.

^{[3]}Let 𝔽q{\mathbb{F}_{q}} be the finite field of order q, where q is an odd prime power.

^{[4]}For $g=12$, we present lower and upper bounds for this invariant when $q\ge 9$ an odd prime power.

^{[5]}We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,\lambda)$ design with $\gcd(k,\lambda)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq A\Gamma L_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$.

^{[6]}Let $${\mathbb {R}}$$ R be the finite non-chain ring $${\mathbb {F}}_{{ q}^{2}}+{v}{\mathbb {F}}_{{ q}^{2}}$$ F q 2 + v F q 2 , where $${v}^{2}={v}$$ v 2 = v and q is an odd prime power.

^{[7]}Let q be an odd prime power, and denote by $${\mathbb {F}}_q$$ the finite field with q elements.

^{[8]}The resulting designs have 2 s n runs with s being any odd prime power and fill some run-size gaps left by existing constructions.

^{[9]}Inspired by results in Deligne--Lusztig theory for classical groups, if $q$ is an odd prime power we propose a set $\operatorname{Irr}(\mathbf{X}(q))$ of `ordinary irreducible characters' associated to the space $\mathbf{X}(q)$ of homotopy fixed points under the unstable Adams operation $\psi^q$.

^{[10]}In this paper, we utilize the decomposition of the defining set and

^{[11]}In this paper, two families of Hermitian dual-containing Bose-Chau- dhuri-Hocquenghem (BCH) codes with length n=a⋅q2+12$n=a\cdot \frac {q^{2}+ 1}{2}$ and n = b (q2 + 1) are studied, where odd a∣(q − 1) for odd prime power q and b∣ (q + 1) for even prime power q, respectively.

^{[12]}We show that under certain technical conditions that simple (q,k,iλmin) balanced incomplete block designs (BIBDs) exist for all allowable values of i, where q is an odd prime power.

^{[13]}

## odd prime p

As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.^{[1]}In this paper, we introduced certain formulas for p-adic valuations of Stirling numbers of the second kind S(n, k) denoted by vp(S(n, k)) for an odd prime p and positive integers k such that n ≥ k.

^{[2]}For an odd prime p and q = pr, this paper deals with LCD codes obtained from cyclic codes of length n over a finite commutative non-chain ring $\mathcal {R}=\mathbb {F}_{q}[u,v]/\langle u^{2}-\alpha u,v^{2}-1, uv-vu\rangle $ where α is a non-zero element in $\mathbb {F}_{q}$.

^{[3]}For any positive integer m > 2 and an odd prime p, let $\mathbb {F}_{p^{m}}$ be the finite field with pm elements and let $ \text {Tr}^{m}_{e}$ be the trace function from $\mathbb {F}_{p^{m}}$ onto $\mathbb {F}_{p^{e}}$ for a divisor e of m.

^{[4]}In this paper we prove that, for an even lattice L, if there exists an odd prime p such that L is local p-maximal and the determinant of L is divisible by p2, then the Eisenstein series of weight 3/2 attached to the discriminant form of L is holomorphic.

^{[5]}$$ For any odd prime p and positive integer n, we establish the new result $$\begin{aligned} \frac{u_{pn}(A,B)-\left( \frac{A^2-4B}{p}\right) u_n(A,B)}{pn}\in {\mathbb {Z}}_p, \end{aligned}$$ where $$\left( \frac{\cdot }{p}\right) $$ is the Legendre symbol and $${\mathbb {Z}}_p$$ is the ring of p-adic integers.

^{[6]}For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p.

^{[7]}Let F q be the finite field of order q , where q is a power of an odd prime p , and p , l are distinct odd primes, and m , n , K are positive integers.

^{[8]}

## odd prime dimension

We introduce a "periodized stationary phase method" to discrete Wigner functions of systems with odd prime dimension and show that the π8 gate is the discrete analog of the Airy function.^{[1]}We prove that the automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields are cyclic, and give a complete list of finite groups that can be realized as such automorphism groups.

^{[2]}

## odd prime order

We use bicyclic units to give an explicit construction of a subgroup of isomorphic to the free product of two free abelian groups of rank two, assuming that G is a finite nilpotent group and it contains an element g of odd prime order such that the subgroup is not normal in G.^{[1]}All parallelisms of \(\mathrm{PG}(3,2)\) and \(\mathrm{PG}(3,3)\) are known and parallelisms of \(\mathrm{PG}(3,4)\) which are invariant under automorphisms of odd prime orders and under the Baer involution have already been classified.

^{[2]}

## odd prime divisor

org/1998/Math/MathML">^{[1]}We introduce admissible collections for a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the Quillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of |G|\lvert G\rvert.

^{[2]}

## odd prime large

org/1998/Math/MathML" id="M1">^{[1]}Let $ p $ be an odd prime large enough.

^{[2]}