Abstract
Let
τ
(
n
) is the number of all divisors of n and
σ
(
n
) is the number of summation of all divisors n, Cavior, presented the number of all subgroups of the dihedral group is equal by
τ
(
n
...) +
σ
(
n
). We in this paper determines a formula for the number of subgroups, normal and cyclic subgroups of the group G =
D
2
n
×
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.
In this paper we find the Circular Divison of The ℱactor Group cℱ(2 2κ×D4, ℤ)/\(\bar{\rho }\)(2 2κ ×D4) when κ is a prime number, where 2 2κ is denoted to Quaternion group of order 4k, such that for ...each positive integer n, there are two generators X and Y for 2 2κ satisfies Q2k={ X i Y j, 0≤ i ≤ 2κ − 1, j=0, 1} which has the following properties{ X 2k=Y 4=I, Y X k Y -1=X -k} and D4 is the Dihedral group of order 8 is generate by a rotation e of order 4 and reflection f of order 2 then 8 elements of D4 can be written as: {I*, e, e2, e3, f, fe, fe2, fe3}.
By using the generalized bilinear operations with a prime number p=3, a (2+1)-dimensional gBK-like equation is introduced. classes of interaction solutions of the (2+1)-dimensional gBK-like equation ...are generated through symbolic computation with maple. Some images were plotted to illustrate the dynamical movement of the solutions with specific values of the involved parameters.
This article considers primes p such that 2kp + 1 is also prime for some natural number k. And, we prove that if 2 is a primitive root of 2kp + 1, then p ≡ k (mod 4). We also present a few more ...observations concerning these primes and the least primitive root of 2kp + 1.
The positive integer points of elliptic curves are very important in the theory of numbers and arithmetic algebra; it has a wide range of applications in cryptography and other fields.The main ...purpose of this paper is to apply elementary methods, the properties of congruence and Legendre symbols, to study the elliptic curve y2 = 7 qx(x2 + 128)and proved that the elliptic curve has at most three integer points when q = 5(mod8) is an odd prime number.
By using elementary methods such as the properties of congruence and Legendre symbols, it is proved that the elliptic curve y2 = 7mx(x2 + 512) has at most 1 positive integer point when q ≡ 5(mod8) is ...an odd prime number.
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