## 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 $$\nu \ge 3$$.

New entanglement-assisted quantum MDS codes with length $$n=\frac{q^2+1}{10\mu }$$

## In this paper, we utilize the decomposition of the defining set and <inline-formula> <tex-math notation="LaTeX">$q^{2}$ </tex-math></inline-formula>-cyclotomic cosets of constacyclic codes with the form <inline-formula> <tex-math notation="LaTeX">$q=\alpha m+t$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">$q=\alpha m+\alpha -t$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n= (q^{2}+1/\alpha)$ </tex-math></inline-formula> to construct some new families of entanglement-assisted quantum MDS codes that satisfy the entanglement-assisted quantum singleton bound, where <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is an odd prime power and <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is a positive integer, while both <inline-formula> <tex-math notation="LaTeX">$\alpha$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> are positive integers such that <inline-formula> <tex-math notation="LaTeX">$\alpha =t^{2}+1$ </tex-math></inline-formula>.

Some New Classes of Entanglement-Assisted Quantum MDS Codes Derived From Constacyclic Codes

## 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.

A Note on Gauss’s Theorem on Primitive Roots

## 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.

Trace representation of the binary pq2-periodic sequences derived from Euler quotients

## 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.

The diophantine equation x2 + paqb = yq

## 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 $9$.

Generalized dihedral CI-groups

## 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.

Paramedial quasigroups of prime and prime square order

## 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.

A geometric approach to Quillen’s conjecture

## 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.

Generators for the Mod-p Cohomology of the Steinberg Summand of Thom Spectra Over $\mathrm {B}(\mathbb {Z}/p)^{n}$-Odd Primary Cases

## In this paper, we completely classify the finite p-groups with $$\nu _c=p$$ or $$p+1$$ for an odd prime number p.

On the Conjugacy Classes of Cyclic Non-normal Subgroups

## Let p be an odd prime number and g < 2 be an integer.

Fast Computation of Hyperelliptic Curve Isogenies in Odd Characteristic

## 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 $$\nu \ge 3$$.

New entanglement-assisted quantum MDS codes with length $$n=\frac{q^2+1}{10\mu }$$

## Let <inline-formula> <tex-math notation="LaTeX">$q = r^{2}$ </tex-math></inline-formula> be an odd prime power.

New Results on Self-Dual Generalized Reed-Solomon Codes

## As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

The Mean Values of Character Sums and Their Applications

## 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.

On the P-Adic Valuations of Stirling Numbers of the Second Kind

## 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.

Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

## 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.

Automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields

## 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.

On free products inside the unit group of integral group rings

## 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.

Parallelisms of \mathrmPG(3, 4) Invariant Under Cyclic Groups of Order 4

## org/1998/Math/MathML"> <mml:mrow> <mml:mo>gcd</mml:mo> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and the odd prime divisors of <jats:italic>m</jats:italic>.

On a bound of Cocke and Venkataraman

## 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.

A geometric approach to Quillen’s conjecture

## org/1998/Math/MathML" id="M1"> <mi>p</mi> [/itex] </jats:inline-formula> be an odd prime large enough.

On Pythagorean Triples and the Primitive Roots Modulo a Prime

## Let $p$ be an odd prime large enough.

Pythagorean triples and quadratic residues modulo an odd prime

10.1142/s0218196721500065

## 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 $X$.

An extension of the Glauberman ZJ-theorem

10.1515/JGTH-2020-0197

## In this paper, we introduce the notion of a quasi-powerful p-group for odd primes p.

Quasi-powerful 𝑝-groups

10.1142/s1793042122500336

## Let [Formula: see text] be an odd prime and [Formula: see text].

On Artin’s conjecture for pairs of diagonal forms

## 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 $\chi$.

A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications

10.1142/S1793042122500154

## Let [Formula: see text] be a positive integer and let [Formula: see text] be an odd prime.

Consecutive square-free numbers and square-free primitive roots

10.1088/1742-6596/2010/1/012068

## 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.

The balanced quaternary sequence of odd period with low autocorrelation

10.12732/ijam.v33i6.4

## Let p be an odd prime.

FULLY-DIVERSE LATTICES FROM RAMIFIED CYCLIC EXTENSIONS OF PRIME DEGREE

10.3390/MATH9040318

## As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

The Mean Values of Character Sums and Their Applications

10.1007/s11139-021-00478-9

## 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 $$d=3$$.

Some q-analogues of supercongruences for truncated $$_3F_2$$ hypergeometric series

10.1007/s00605-021-01587-9

## org/1998/Math/MathML"> <mml:mrow> <mml:mo>gcd</mml:mo> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and the odd prime divisors of <jats:italic>m</jats:italic>.

On a bound of Cocke and Venkataraman

10.1007/s12095-021-00530-x

## 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.

Differential and boomerang spectrums of some power permutations

10.1142/S1793830921501135

## 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.

Skew cyclic codes over 𝔽q[u,v,w]/〈u2 − 1,v2 − 1, w2 − 1,uv − vu,vw − wv,wu − uw〉

10.1109/tit.2021.3096934

## Let <inline-formula> <tex-math notation="LaTeX">$q = r^{2}$ </tex-math></inline-formula> be an odd prime power.

New Results on Self-Dual Generalized Reed-Solomon Codes

10.1016/J.JNT.2021.06.030

## 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.

Anticyclotomic μ-invariants of residually reducible Galois representations

10.1007/S13226-021-00001-2

## Let $p$ be an odd prime, and let $m$ be a positive integer satisfying $p^m \equiv 3~(\text{mod }4). A class of constacyclic codes over $${\mathbb{F}}_{p^m}[u]/\left\langle u^2\right\rangle$$ ## In this paper, we show that$f(k)$exists if and only if$k$is an odd prime. Circular coloring and fractional coloring in planar graphs 10.1007/S41980-020-00502-6 ## In this paper, we completely classify the finite p-groups with $$\nu _c=p$$ or $$p+1$$ for an odd prime number p. On the Conjugacy Classes of Cyclic Non-normal Subgroups 10.1016/j.ffa.2021.101897 ## In this paper, for an odd prime$p$, by extending Li et al. Complete weight enumerators for several classes of two-weight and three-weight linear codes 10.3329/JSR.V13I1.47721 ## 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. Some Results on the Sandor-Smarandache Function 10.1007/S11139-021-00466-Z ## 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. Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions 10.1017/S0004972721000563 ## 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. A NOTE ON GROUP RINGS WITH TRIVIAL UNITS 10.1145/3452143.3465523 ## Let p be an odd prime number and g < 2 be an integer. Fast Computation of Hyperelliptic Curve Isogenies in Odd Characteristic 10.1155/2021/7634728 ## org/1998/Math/MathML" id="M1"> <mi>p</mi> [/itex] </jats:inline-formula> be an odd prime large enough. On Pythagorean Triples and the Primitive Roots Modulo a Prime 10.25394/PGS.15073497.V1 ## 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. Finite quotients of triangle groups 10.37256/CM.212021717 ## 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. On the P-Adic Valuations of Stirling Numbers of the Second Kind 10.1007/s10114-021-9509-3 ## Let p be an odd prime, and let k be a nonzero nature number. Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup 10.1155/2021/5560902 ## 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. On the Hybrid Fourth Power Mean Involving Legendre’s Symbol and One Kind Two-Term Exponential Sums 10.7546/nntdm.2021.27.3.104-112 ## 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). Heilbronn-like sums and their properties 10.1515/forum-2020-0144 ## Let 𝔽q{\mathbb{F}_{q}} be the finite field of order q, where q is an odd prime power. Incidences between Euclidean spaces over finite fields 10.1016/j.dam.2020.10.009 ## For$g=12$, we present lower and upper bounds for this invariant when$q\ge 9$an odd prime power. On the packing chromatic number of Moore graphs 10.3934/amc.2020049 ## 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 $\end{document}. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes 10.1007/s10801-021-01068-0 ## We, in particular, prove that if$\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)$. A classification of flag-transitive block designs 10.3934/amc.2021028 ## <p style='text-indent:20px;'>For an odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_{p^m} $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M5">\begin{document}$ p^{m} $\end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id="M6">\begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document}</tex-math></inline-formula>. New quantum codes from skew constacyclic codes 10.1016/J.JNT.2021.03.012 ## Let p be an odd prime number. Root numbers for the Jacobian varieties of Fermat curves 10.1007/s00222-021-01057-x ## For odd primes$p$, we construct closed locally CAT(0) manifolds with nonzero mod$p$homology growth outside the middle dimension. Mod p and torsion homology growth in nonpositive curvature 10.3934/math.2021638 ## We consider the calculation problem of one kind hybrid power mean involving the character sums of polynomials and two-term exponential sums modulo$ p $, an odd prime, and use the analytic method and the properties of classical Gauss sums to give some identities and asymptotic formulas for them. The hybrid power mean of some special character sums of polynomials and two-term exponential sums modulo$ p $10.1007/s12095-021-00516-9 ## For an odd prime p and q = pr, this paper deals with LCD codes obtained from cyclic codes of length n over a finite commutative non-chain ring$\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}$. Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring 10.1088/1742-6596/1818/1/012081 ## We in this paper determines a formula for the number of subgroups, normal and cyclic subgroups of the group G = D 2n × C p = 〈a, b, c|a n = b 2 = c p , b a b = a −1, [a, c] = [b, c] = 1〉, where p is an odd prime number. Computing the Certain subgroups of the group D 2n × C p , p is an Odd Prime Number 10.3836/TJM/1502179326 ## Let$p=2^{e+1}q+1$be an odd prime number with$2 \nmid q$. On the Class Group of an Imaginary Cyclic Field of Conductor$8p$and$2$-power Degree 10.22108/TOC.2020.123692.1740 ## Let$G = PSL_{2}(q)$‎, ‎where$q$is a power of an odd prime‎. Symmetric$1$-designs from$PSL_{2}(q),$for$q$a power of an odd prime 10.1007/s12346-021-00453-1 ## 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. On the Lefschetz Zeta Function for a Class of Toral Maps 10.1007/S40316-021-00168-4 ## 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. A short note on inadmissible coefficients of weight 2 and $$2k+1$$ newforms 10.1007/S12095-021-00496-W ## 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}}$for a divisor e of m. A class of linear codes with their complete weight enumerators over finite fields ## In the present paper, we prove that, for an odd prime number$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. A note on the existence of Tango curves 10.1007/S00025-021-01389-3 ## 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. Proof of Two Congruences Concerning Legendre Polynomials 10.1080/00927872.2021.1879106 ## Let p be an odd prime and the integral group ring of the dihedral group of order 2p. Syzygy modules for dihedral groups 10.24143/2072-9502-2021-1-70-79 ## Sequences are formed on the basis of new generalized cyclotomy modulo equal to the degree of an odd prime. Analysis of linear complexity of generalized cyclotomic Q-ary sequences of pn period 10.1134/S0037446621020178 ## 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 $. On the Local Case in the Aschbacher Theorem for Symplectic and Orthogonal Groups 10.1017/s0305004121000657 ## Let A be an abelian variety defined over a number field k, let p be an odd prime number and let$F/k$be a cyclic extension of p-power degree. Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III 10.1016/j.disc.2020.112189 ## Let p be an odd prime, and k an integer such that k ∣ ( p − 1 ). A class of skew cyclic codes and application in quantum codes construction 10.1007/s40840-021-01157-0 ## In this paper, on the one hand, using some properties on the representation of integers by binary quadratic primitive forms, we give a new and elementary proof for Dem’janenko’s result; on the other hand, using the Baker method and its p-adic form, we prove that if $$n>1$$ , $$f=g+1$$ and $$g=2^r$$ , where r is a positive integer with $$r\ge 80$$ and $$r+1$$ is an odd prime, then $$(*)$$ has only the positive integer solution $$(x,y,z)=(2,2,2)$$. Dem’janenko’s Theorem on Jeśmanowicz’ Conjecture Concerning Pythagorean Triples Revisited 10.1007/s11128-021-03052-w ## 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. Quantum codes from Hermitian dual-containing constacyclic codes over $${\mathbb {F}}_{q^{2}}+{v}{\mathbb {F}}_{q^{2}}$$ F q 10.22044/JAS.2020.8875.1431 ## 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. ON THE CLASS OF ARRAY-BASED APM-LDPC CODES 10.22331/q-2021-07-05-494 ## 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. Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits ## the homology of the image of the restriction Res(Spn ,E n ) : H∗(Spn ;Fp ) → H∗(En ;Fp ) with the differential to be the Milnor operation Q j , for p an odd prime and for any n, j. The mod p Margolis homology of the Dickson–Mùi algebra 10.35940/ijbsac.d0216.083421 ## In this paper, the researcher has examined the solutions of Diophantine equation (𝑴𝒑 − 𝟐) 𝒙 + (𝑴𝒑 + 𝟐) 𝒚 = 𝒛 𝟐 where 𝑴𝒑 is a Mersenne Prime and p is an odd prime whereas x, y and z are nonnegative integers. On the Solutions of Diophantine Equation (Mp − 2) x + (Mp + 2) y = z 2 where Mp is Mersenne Prime 10.3934/amc.2020044 ## Let \begin{document}$ {\mathbb F}_q $\end{document} be the finite field with \begin{document}$ q = p^m $\end{document} elements, where \begin{document}$ p $\end{document} is an odd prime and \begin{document}$ m $\end{document} is a positive integer. A class of linear codes and their complete weight enumerators 10.1016/j.ipl.2020.106078 ## Our results confirm that Kurepa's left factorial conjecture is still an open problem, as they show that there are no odd primes$p<2^{40}$such that$p$divides$!p$. Improved algorithms for left factorial residues 10.4134/BKMS.B200375 ## 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. A note on vector-valued Eisenstein series of weight$3/2$10.1007/S10474-020-01123-5 ## Alon [1] proved that if $$p$$ is an odd prime, $$1\le n < p$$ and $$a_1,\ldots,a_n$$ are distinct elements in $$Z_p$$ and $$b_1,\ldots,b_n$$ are arbitrary elements in $$Z_p$$ then there exists a permutation of $$\sigma$$ of the indices $$1,\ldots,n$$ such that the elements $$a_1+b_{\sigma(1)},\ldots,a_n+b_{\sigma(n)}$$ are distinct. A Snevily-type inequality for multisets 10.1093/qmath/haab023 ## We also present a list of necessary conditions for a partition to have Schaper number at least three for odd primes and a conjecture on the sufficiency of these conditions. On The Schaper Numbers of Partitions 10.1155/2021/9570350 ## org/1998/Math/MathML" id="M2"> <mi>p</mi> [/itex] </jats:inline-formula> be an odd prime and let <jats:inline-formula> <math xmlns="http://www. The p -Adic Valuations of Sums of Binomial Coefficients 10.1016/j.jnt.2021.08.007 ## Let$p$be an odd prime and consider two elliptic curves$E_1, E_2$with good, ordinary reduction at primes above$p$and equivalent mod-$p$Galois representations. Multiplicities in Selmer groups and root numbers of Artin twists ## All non-trivial point and block-primitive 1-(v, k, k) designs 𝓓 that admit the group G = PGL2(q), where q is a power of an odd prime, as a permutation group of automorphisms are determined. Symmetric 1-designs from PGL2(q), for q an odd prime power 10.1007/S12190-021-01499-9 ## Let q be an odd prime power, and denote by $${\mathbb {F}}_q$$ the finite field with q elements. Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$ 10.1016/j.ejc.2021.103404 ## Let$p$be an odd prime. On the Baer-Lovász-Tutte construction of groups from graphs: Isomorphism types and homomorphism notions 10.3934/math.2022057 ## Let$ p be an odd prime large enough. Pythagorean triples and quadratic residues modulo an odd prime 10.1007/S00200-021-00507-6 ## Let p be any odd prime number and let m, k be arbitrary positive integers. An explicit expression for all distinct self-dual cyclic codes of length $$p^k$$ over Galois ring $$\mathrm{GR}(p^2,m)$$ 10.1007/s00200-021-00492-w ## 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. A note on “H. Q. Dinh et al., Hamming distance of repeated-root constacyclic codes of length $$2p^{s}$$ 2 p s 10.1007/S00605-021-01567-Z ## $$For any odd prime p and positive integer n, we establish the new result$$\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$$is the ring of p-adic integers. Supercongruences involving Lucas sequences 10.1142/s0218196721500077 ## For any odd prime p, we give an example of a locally finite p-group G containing a left 3-Engel element x where \langle x \rangle^G is not nilpotent. Locally finite p-groups with a left 3-Engel element whose normal closure is not nilpotent 10.1007/s13226-021-00080-1 ## In this paper, we consider the title simultaneous Pell equations, where$$a\ge 2,b \ge 1$$a ≥ 2 , b ≥ 1 are positive integers and p is an odd prime. On the system of Pell equations$$x^2-(a^2b^2 {\pm } a)y^2=1$$x 2 - ( 10.1007/s00145-021-09401-3 ## We give an algorithm to compute$$(\ell ,\ell ,\ell )$$( ℓ , ℓ , ℓ ) -isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves over a finite field of characteristic different from 2 in time$$\tilde{O}(\ell ^3)$$O ~ ( ℓ 3 ) , where$$\ell $$ℓ is an odd prime which is coprime to the characteristic. Translating the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves with$$(\ell ,\ell ,\ell )\$ ( ℓ , ℓ , ℓ ) -Isogenies

10.15408/INPRIME.V3I1.19670

## 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.

Some Results of The Coprime Graph of a Generalized Quaternion Group Q_4n

10.1016/j.jnt.2020.10.004

## In this note, we confirm a conjectural formula on the number of representations of the square of an odd prime by a sum of an odd number of squares, i.

Proof of a conjecture of Cooper

## We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups.

The fields of values of characters of degree not divisible by p