This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together wi...This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.展开更多
Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizatio...Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, <img alt="" src="Edit_e97d0c64-3683-4a75-9d26-4b371c2be41e.bmp" /> where P<em><sub>P </sub></em>denotes the set of positive integers which are prime to <em>p</em>. And we establish the combinational congruences involving alternating harmonic sums for positive integer <em>n</em>=3,4,5.展开更多
This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1（mod 4）or a〉1then [3/4pa]∑...This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1（mod 4）or a〉1then [3/4pa]∑k=0≡（2/pa）（mod p^2）where（—）denotes the Jacobi symbol.This confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that p-1∑k=1 Lk/k^2≡0（mod p） provided p〉5.where the Lucas numbers Lo,L1,L2,...are defined by L_0=2,L1=1 and Ln＋1=Ln＋Ln-l（n=1,2,3,...）.The third theorem states that if p=5 then Fp^a-（p^a/5）mod p^3 can be determined in the following way： p^a-1∑k=0（-1）^k（2k k）≡（p^a/5）（1-2F p^a-（pa/5））（mod p^3）which appeared as a conjecture in a paper of Sun and Tauraso in 2010.展开更多
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
In this paper, the additive equations of the type α_1λ_1~k+ … +α_sλ_s^k = 0 are studied, α_i'sbeing integers of an algebraic number field K of degree n. The main result is as follows: Ifs≥(2k)^(n+1) (or s≥...In this paper, the additive equations of the type α_1λ_1~k+ … +α_sλ_s^k = 0 are studied, α_i'sbeing integers of an algebraic number field K of degree n. The main result is as follows: Ifs≥(2k)^(n+1) (or s≥cknlogk for 2 + k), the equation is solved nontrivially in any β-adic field,where β is a prime ideal of K.展开更多
设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1...设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1时,Diophantine方程x^3-1=13qy^2仅有整数解(x,y)=(1,0).展开更多
文摘This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.
文摘Let <em>p</em> be an odd prime, the harmonic congruence such as <img alt="" src="Edit_843b278d-d88a-45d3-a136-c30e6becf142.bmp" />, and many different variations and generalizations have been studied intensively. In this note, we consider the congruences involving the combination of alternating harmonic sums, <img alt="" src="Edit_e97d0c64-3683-4a75-9d26-4b371c2be41e.bmp" /> where P<em><sub>P </sub></em>denotes the set of positive integers which are prime to <em>p</em>. And we establish the combinational congruences involving alternating harmonic sums for positive integer <em>n</em>=3,4,5.
基金supported by National Natural Science Foundation of China(Grant Nos.10901078 and 11171140)
文摘This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1（mod 4）or a〉1then [3/4pa]∑k=0≡（2/pa）（mod p^2）where（—）denotes the Jacobi symbol.This confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that p-1∑k=1 Lk/k^2≡0（mod p） provided p〉5.where the Lucas numbers Lo,L1,L2,...are defined by L_0=2,L1=1 and Ln＋1=Ln＋Ln-l（n=1,2,3,...）.The third theorem states that if p=5 then Fp^a-（p^a/5）mod p^3 can be determined in the following way： p^a-1∑k=0（-1）^k（2k k）≡（p^a/5）（1-2F p^a-（pa/5））（mod p^3）which appeared as a conjecture in a paper of Sun and Tauraso in 2010.
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.
文摘In this paper, the additive equations of the type α_1λ_1~k+ … +α_sλ_s^k = 0 are studied, α_i'sbeing integers of an algebraic number field K of degree n. The main result is as follows: Ifs≥(2k)^(n+1) (or s≥cknlogk for 2 + k), the equation is solved nontrivially in any β-adic field,where β is a prime ideal of K.
文摘设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1时,Diophantine方程x^3-1=13qy^2仅有整数解(x,y)=(1,0).
