If odd primes p and q are such that p≡3(mod 16) and q≡15(mod 16) or p≡11(mod 16) and q≡7(mod 16) in E_(-pq):y^2=x^3-pqx then, rank 2 can be deduced([2] and [4]). It is trivial that we pursue primes p and q as complex as possible. The primes of the forms p=( ) u^4+( ) v^4+( ) w^4+( ) u^2 v^2+( ) u^2 w^2+( ) v^2 w^2and q=( ) 〖'u〗^4+( ) 〖'v〗^4+( ) 〖'w〗^4+( )^' u^2 v^2+( )^' u^2 w^2+( )^' v^2 w^2 in [5] was the beginning taking more numbers of terms than p=Hu^4+Iu^2 v^2+Kv^4 and q=H〖'u〗^4+I^' u^2 v^2+K^' v^4. In [10], the primes p and q are composed of 22 variables and 253 terms where variables are A and B and C and D and E and F and G and H and I and J and K and L and O and P and Q and R and S and T and U and V and W and Z. In this article, we add more variables to [10].
An element of a unital ring is called strongly not clean if all of its unit multiples fail to be clean and very strongly not clean if all its nonzero products with arbitrary (nonzero) elements fail to be clean. In this paper, we investigate such elements, with a particular focus on matrices over commutative rings. In our main result we show that a $2\times 2$ matrix over any Bézout domain with almost stable range 1 is vsn-clean iff its entries are not coprime, excepting $2I_{2}$.
We appoint that E_pq is an elliptic curve y^2=x^3+pqx with distinct odd primes p and q. If p is supposed as the form p≡1(mod 8) then, there can be given rank at least 2 according to defining prime q. For taking at least 2, generally primes are gotten as p=Hu^4+Iu^2 v^2+Kv^4 and q=H'u^4+I'u^2 v^2+K'v^4⋯⋯(BB) . In this article, we will treat rank of curve E_pq where primes are composed of more terms than (BB).
Set E_(-pqs) as an elliptic curve y^2=x^3-pqsx with different odd primes p and q and s then, we regard rank of this curve.
Assign E_(p(p-2)) as an elliptic curve y^2=x^3+p(p-2)x with twin primes then, we shall treat the rank of this curve and if E_(-2p) is gotten as an elliptic curve y^2=x^3-2px then, we will calculate the rank and compare the results with previous curves.
In curve E_(-pq): y^2=x^3-pqx with distinct primes p and q we can attain rank 2. The primes are gotten as various forms. The forms p=Hu^4+Iu^2 v^2+Kv^2 and 〖q=H^' u〗^4+I^' u^2 v^2+K^' v^4 are beginning of the forms. We pursue as many terms as possible in primes. In this article, we consider rank of curve with this point.
We prove that $\mathbb{R}$, $\mathbb{C}$, $\stackrel{\star}{\mathbb{C}}$, $\mathbb{H}$, $\stackrel{\star}{\mathbb{H}}$, $\mathbb{O}$ and $\stackrel{\star}{\mathbb{O}}$ are the only third-power associative absolute valued real algebras with a nonzero weak central element. We show also that if $A$ is a third-power associative absolute valued real algebra with a nonzero alternative element, then $A$ is power-associative, and isomorphic to $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ or $\mathbb{O}$.
We will numerate the rank of elliptic curve E_(-2p): y^2=x^3-2px with prime p as 〖p=As〗^4+Bt^4+Cu^4+Dv^4+Fw^4+Gs^2 t^2+Hs^2 u^2+〖Is〗^2 v^2+Js^2 w^2+Kt^2 u^2+Lt^2 v^2+Rt^2 w^2+Su^2 v^2+〖Uu〗^2 w^2+Vv^2 w^2 and submit examples of the result.
Define E_(-2p) as an elliptic curve y^2=x^3-2px then, we will research the rank of it and submit several examples.
Take E_(-p(p-2)) as an elliptic curve y^2=x^3-p(p-2)x with twin primes p and p-2 then, we shall investigate the rank of it. Denote E_(-2p) and E_(-4p) as elliptic curves y^2=x^3-2px and y^2=x^3-4px then, we will research the ranks and compare the results with previous curve.
Denote E_pq as an elliptic curve y^2=x^3+pqx with distinct odd primes p and q then, we shall compute the rank of curve where p and q are composed of more than 10 variables and 15 terms.
Suppose that P_i is strongly irreducible ideal which contains idealIof commutative ring R with identity. Then, we investigate whether the union of √(P_i )/I in R/I is strongly irreducible or not. In addition, we will treat that whether intersection of strongly irreducible is strongly irreducible or not.
Take elliptic curves E_(-pq) and E_(-2pq) as y^2=x^3-pqx and y^2=x^3-2pqx with distinct odd primes p and q then, we will compare the ranks of curves according to each condition.
We denote E_(-2p) as an elliptic curve y^2=x^3-2px with prime p=〖Hu〗^4±Iu^2 v^2+Kv^4 then, we research the rank and submit examples.
This article addresses a common misconception about combining inequalities. Inequality is used in algebra to show magnitude comparison, as well as in set theory and mathematical programming to represent a set of values satisfying inequality. It is a widespread misconception that two inequalities can be combined under any circumstances. In this article, we prove that when inequality represents a set of values, combining inequalities would fail to maintain the set they represent and would result in a larger set. Therefore, even though it is valid when inequality is used to represent relation of magnitude comparison, combining inequalities is not valid when inequality is used to represent a set of values. Six examples are diagnosed in applications of solving simultaneous equations and constructing constraints in linear and integer programming, and pitfalls due to invalidly combining inequalities are indicated in those examples.
In this paper, we study the homogeneous structures of pseudo-Riemannian Lie groups of signature $(2,2)$. This study allows to classify the cyclic Lie groups of the mentioned signature. We also study the cyclic pseudo-Riemannian homogeneous $4$-manifolds with non-trivial isotropy.
In form E_(-2p):y^2=x^3-2px we can obtain rank at least 2 when prime is p≡1(mod 8). In this case, generally the form is given as p=Hu^8+Iu^4 v^4+Kv^8. That is, the numbers of terms are 3. Our concern is whether it is possible that we attain more terms in primes. In this article, we will treat about this point.
This paper investigates a novel characterization method for the sporadic simple group $B$ (Baby Monster Group). Using the combination of the order component set $OC(G)$ of a finite group and the set $\pi_{p_m}(G)$ of the orders of centralizers of elements of highest order within the group, we proved: a finite group $G$ is isomorphic to the sporadic simple group $B$ if and only if the following two conditions hold: (1) $G$ shares the same largest order component $m_1(G)$ as B; (2) The sets $\pi_{p_m}(G)$ and $\pi_{p_m}(B)$ of the orders of centralizers of their respective highest-order elements are identical. By analyzing the prime graph structure, excluding the possibility of Frobenius groups and 2-Frobenius groups, and utilizing the Classification Theorem of Finite Simple Groups, we ultimately establish the sufficiency and necessity of these characterizing conditions.
This paper gives the notion of orthogonality between the left reverse derivation and symmetric left reverse biderivation of a semiprime ring. We prove that if R is a semiprime ring, B is a reverse biderivation and d is a derivation of R are orthogonal if and only if any one of the following equivalent conditions hold for every x,y,z∈R. (i)B(x,y)d(z)+d(x)B(z,y)=0 (ii) dB=0 (iii) d(x)B(x,y)=0 (iv) dB is a biderivation.
