The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infini...The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.展开更多
This paper is to study the distribution property of Lehmer DH number by use the estimation of general Kloostermann sum and the estimation of trigonometric sum.
We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive A-roots in any interval I of length ｜I｜ 〉〉 q^1/2＋ε Similar resu...We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive A-roots in any interval I of length ｜I｜ 〉〉 q^1/2＋ε Similar results are obtained for the distribution of ordered pairs （x, x^-1） with x a primitive λ-root, and for the number of primitive A-roots satisfying inequalities such as ｜x - x^-1｜≤ B.展开更多
It is proved that among the regular polygons with prime edges, only the regular polygons with Fermat prime edges are constructable with compass and straightedge. As 17 is a Fermat prime number, the construction of hep...It is proved that among the regular polygons with prime edges, only the regular polygons with Fermat prime edges are constructable with compass and straightedge. As 17 is a Fermat prime number, the construction of heptadecagon has been discussing all the time. Many different construction methods are proposed although they are based on the same theory. Simplification of the construction is still a sensible problem. Here, we propose a simple method for constructing regular heptadecagon with the fewest steps. The accumulation of construction errors is also avoided. This method is more applicable than the previous construction.展开更多
Median noise barriers, like parallel noise barriers, can be employed to reduce the impact of traffic on roadside communities via the direct propagation path. The performance of different shapes of median barriers was ...Median noise barriers, like parallel noise barriers, can be employed to reduce the impact of traffic on roadside communities via the direct propagation path. The performance of different shapes of median barriers was compared using reactive and passive surfaces and a 2D boundary element method (BEM). In the case of reactive surfaces, quadratic residue diffusers (QRDs) and primitive root diffusers (PRDs) were used on the top and stem surfaces of median barriers. To introduce passive barriers, two different absorbent materials including fibrous material and a grass surface with flow resistivity of 20 000 and 2500 kg/(s·m2), respectively, were similarly applied. The effect of thin absorptive barriers was similar at lower frequencies and better at mid and high frequencies to that of their equivalent rigid barriers. More improvement was achieved by covering the top surface of thick barriers with grass rather than with fibrous material. The performance of QRD and PRD barriers where the diffuser was located on the top surface was more frequency dependent than that of barriers coated with fibrous material. A comparison of the average A-weighted insertion loss in the thick barriers showed that the greatest improvement (2.59 dB (A)) was achieved using a barrier of 30-cm thickness covered with grass.展开更多
In this paper,we obtain the factorization of direct production and order of group GL（n,Z_m） in a simple method.Then we generalize some properties of GL（2,Z_p） proposed by Huppert,and prove that the group GL（2,Z_...In this paper,we obtain the factorization of direct production and order of group GL（n,Z_m） in a simple method.Then we generalize some properties of GL（2,Z_p） proposed by Huppert,and prove that the group GL（2,Z_z^y） is solvable.We also prove that group GL（n,Z_p）is solvable if and only if GL（n,Z_p） is solvable,and list the generators of groups GL（n,Z_p） and SL（n,Z_p）.At last,we prove that PSL（2,Z_p）（ p〉3） and PSL（n,Z_p） （ n〉3） are simple.展开更多
文摘The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.
基金Supported by the Education Commission Science Foundation of Shaanxi(OOJK123)
文摘This paper is to study the distribution property of Lehmer DH number by use the estimation of general Kloostermann sum and the estimation of trigonometric sum.
基金Project supported by the National Natural Science Foundation of China (No.19625102)the 973 Project of the Ministry of Science and Technology of China.
文摘We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive A-roots in any interval I of length ｜I｜ 〉〉 q^1/2＋ε Similar results are obtained for the distribution of ordered pairs （x, x^-1） with x a primitive λ-root, and for the number of primitive A-roots satisfying inequalities such as ｜x - x^-1｜≤ B.
文摘It is proved that among the regular polygons with prime edges, only the regular polygons with Fermat prime edges are constructable with compass and straightedge. As 17 is a Fermat prime number, the construction of heptadecagon has been discussing all the time. Many different construction methods are proposed although they are based on the same theory. Simplification of the construction is still a sensible problem. Here, we propose a simple method for constructing regular heptadecagon with the fewest steps. The accumulation of construction errors is also avoided. This method is more applicable than the previous construction.
基金Project (No.8707) supported by the Tehran University of Medical Sciences,Iran
文摘Median noise barriers, like parallel noise barriers, can be employed to reduce the impact of traffic on roadside communities via the direct propagation path. The performance of different shapes of median barriers was compared using reactive and passive surfaces and a 2D boundary element method (BEM). In the case of reactive surfaces, quadratic residue diffusers (QRDs) and primitive root diffusers (PRDs) were used on the top and stem surfaces of median barriers. To introduce passive barriers, two different absorbent materials including fibrous material and a grass surface with flow resistivity of 20 000 and 2500 kg/(s·m2), respectively, were similarly applied. The effect of thin absorptive barriers was similar at lower frequencies and better at mid and high frequencies to that of their equivalent rigid barriers. More improvement was achieved by covering the top surface of thick barriers with grass rather than with fibrous material. The performance of QRD and PRD barriers where the diffuser was located on the top surface was more frequency dependent than that of barriers coated with fibrous material. A comparison of the average A-weighted insertion loss in the thick barriers showed that the greatest improvement (2.59 dB (A)) was achieved using a barrier of 30-cm thickness covered with grass.
文摘In this paper,we obtain the factorization of direct production and order of group GL（n,Z_m） in a simple method.Then we generalize some properties of GL（2,Z_p） proposed by Huppert,and prove that the group GL（2,Z_z^y） is solvable.We also prove that group GL（n,Z_p）is solvable if and only if GL（n,Z_p） is solvable,and list the generators of groups GL（n,Z_p） and SL（n,Z_p）.At last,we prove that PSL（2,Z_p）（ p〉3） and PSL（n,Z_p） （ n〉3） are simple.
