Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and...Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that ∑k=0^p-1（k^2k/2k）≡（-1）^（p-1）/2-p^2Ep-3（modp^3） ∑k=1^（p-1）/2（k^2k）/k≡（-1）^（p＋1）/2 8/3pEp-3（mod p^2）,∑k=0^（p-1）/2（k^2k）^2/16k≡（-1）^（p-1）/2＋p^2Ep-3（mod p^3）,where E0, E1, E2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π-2 and the constant K ：= ∑k=1^∞（k/3）/k^2 （with （-） the Jacobi symbol）, two of which are ∑k=1^∞（10k-3）8k/k2（k^2k）^2（k^3k）=π^2/2and ∑k=1^∞（15k-4）（-27）^k-1/k^3（k^2k）^2（k^3k）=K.展开更多
Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.展开更多
Let 2m>2, m∈ℤ, be the given even number of the Strong Goldbach Conjecture Problem. Then, m can be called the median of the problem. So, all Goldbach partitions (p,q)exist a relationship, p=m−dand q=m+d, where p≤q...Let 2m>2, m∈ℤ, be the given even number of the Strong Goldbach Conjecture Problem. Then, m can be called the median of the problem. So, all Goldbach partitions (p,q)exist a relationship, p=m−dand q=m+d, where p≤qand d is the distance from m to either p or q. Now we denote the finite feasible solutions of the problem as S(2m)={ (2,2m−2),(3,2m−3),⋅⋅⋅,(m,m) }. If we utilize the Eratosthenes sieve principle to efface those false objects from set S(2m)in pistages, where pi∈P, pi≤2m, then all optimal solutions should be found. The Strong Goldbach Conjecture is true since we proved that at least one optimal solution must exist to the problem. The Weak Goldbach Conjecture is true since it is a special case of the Strong Goldbach Conjecture. Therefore, the Goldbach Conjecture is true.展开更多
Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0...Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0【k【pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.展开更多
We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient p...We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.展开更多
The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbe...The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.展开更多
Let a and b be fixed positive integers.In this paper,using some elementary methods,we study the diophantine equation（a^m-1）（b^n-1）= x^2.For example,we prove that if a ≡ 2（mod 6）,b ≡ 3（mod 12）,then（a^n-1）（...Let a and b be fixed positive integers.In this paper,using some elementary methods,we study the diophantine equation（a^m-1）（b^n-1）= x^2.For example,we prove that if a ≡ 2（mod 6）,b ≡ 3（mod 12）,then（a^n-1）（b^m-1）= x^2 has no solutions in positive integers n,m and x.展开更多
Let S be an ideal nil-extension of a completely regular semigroup K by a nil semigroup Q with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a cong...Let S be an ideal nil-extension of a completely regular semigroup K by a nil semigroup Q with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a congruence on K respectively. It is proved that every congruence on S can be uniquely respresented by an admissible congruence pair (δ,ω) of S. Suppose that ρ K denotes the Rees congruence induced by the ideal K of S. Then it is shown that for any congruence σ on S,a mapping Γ:σ|→(σ Q,σ K) is an order-preserving bijection from the set of all congruences on S onto the set of all admissible congruence pairs of S,where σ K is the restriction of σ to K and σ Q=(σ∨ρ K)/ρ K. Moreover,the lattice of congruences of S is also discussed. As a special case,every congruence on completely Archimedean semigroups S is described by an admissible quarterple of S. The following question is asked: Is the lattice of congruences of the completely Archimedean semigroup a semimodular lattice?展开更多
In 2006, Sanwong and Sullivan described the maximal congruences on the semigroup N consisting of all non-negative integers under standard multiplication, and on the semigroup T（X） consisting of all total transformat...In 2006, Sanwong and Sullivan described the maximal congruences on the semigroup N consisting of all non-negative integers under standard multiplication, and on the semigroup T（X） consisting of all total transformations of an infinite set X under composition. Here, we determine all maximal congruences on the semigroup Zn under multiplication modulo n. And, when Y lohtain in X, we do the same for the semigroup T（X, Y） consisting of all elements of T（X） whose range is contained in Y. We also characterise the minimal congruences on T（X. Y）.展开更多
设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1...设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1 k Hk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0 −k a a−1 k Hk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0 −k a a−1 k H(2)k 2k+1≡Bp−2(a)(mod p)其中p表示满足a≤r(mod p)的最小非负整数r,Bn(x)与En(x)分别表示次数为n的伯努利多项式与欧拉多项式.展开更多
Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then...Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.展开更多
基金supported by the National Natural Science Foundation of China(GrantNo.10871087)the Overseas Cooperation Fund of China(Grant No.10928101)
文摘Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that ∑k=0^p-1（k^2k/2k）≡（-1）^（p-1）/2-p^2Ep-3（modp^3） ∑k=1^（p-1）/2（k^2k）/k≡（-1）^（p＋1）/2 8/3pEp-3（mod p^2）,∑k=0^（p-1）/2（k^2k）^2/16k≡（-1）^（p-1）/2＋p^2Ep-3（mod p^3）,where E0, E1, E2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π-2 and the constant K ：= ∑k=1^∞（k/3）/k^2 （with （-） the Jacobi symbol）, two of which are ∑k=1^∞（10k-3）8k/k2（k^2k）^2（k^3k）=π^2/2and ∑k=1^∞（15k-4）（-27）^k-1/k^3（k^2k）^2（k^3k）=K.
基金supported in part by National Natural Science Foundation of China(Grant No.10701048)Natural Science Foundation of Shandong Province (Grant No.ZR2009AM007)+2 种基金Independent Innovation Foundation of Shandong Universitysupported in part by National Basic Research Program of China (973 Program) (Grant No.2007CB807902)National Natural Science Foundation of China (Grant No.10601034)
文摘Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.
文摘Let 2m>2, m∈ℤ, be the given even number of the Strong Goldbach Conjecture Problem. Then, m can be called the median of the problem. So, all Goldbach partitions (p,q)exist a relationship, p=m−dand q=m+d, where p≤qand d is the distance from m to either p or q. Now we denote the finite feasible solutions of the problem as S(2m)={ (2,2m−2),(3,2m−3),⋅⋅⋅,(m,m) }. If we utilize the Eratosthenes sieve principle to efface those false objects from set S(2m)in pistages, where pi∈P, pi≤2m, then all optimal solutions should be found. The Strong Goldbach Conjecture is true since we proved that at least one optimal solution must exist to the problem. The Weak Goldbach Conjecture is true since it is a special case of the Strong Goldbach Conjecture. Therefore, the Goldbach Conjecture is true.
基金supported by National Natural Science Foundation of China (Grant No.10871087)the Overseas Cooperation Fund of China (Grant No.10928101)
文摘Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0【k【pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.
文摘We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.
基金the Guangdong Provincial Natural Science Foundation (No.05005928)the National Natural Science Foundation (No.10671155) of P.R.China
文摘The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.
基金Supported by the National Natural Science Foundation of China (Grant No.10901002)
文摘Let a and b be fixed positive integers.In this paper,using some elementary methods,we study the diophantine equation（a^m-1）（b^n-1）= x^2.For example,we prove that if a ≡ 2（mod 6）,b ≡ 3（mod 12）,then（a^n-1）（b^m-1）= x^2 has no solutions in positive integers n,m and x.
文摘Let S be an ideal nil-extension of a completely regular semigroup K by a nil semigroup Q with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a congruence on K respectively. It is proved that every congruence on S can be uniquely respresented by an admissible congruence pair (δ,ω) of S. Suppose that ρ K denotes the Rees congruence induced by the ideal K of S. Then it is shown that for any congruence σ on S,a mapping Γ:σ|→(σ Q,σ K) is an order-preserving bijection from the set of all congruences on S onto the set of all admissible congruence pairs of S,where σ K is the restriction of σ to K and σ Q=(σ∨ρ K)/ρ K. Moreover,the lattice of congruences of S is also discussed. As a special case,every congruence on completely Archimedean semigroups S is described by an admissible quarterple of S. The following question is asked: Is the lattice of congruences of the completely Archimedean semigroup a semimodular lattice?
文摘In 2006, Sanwong and Sullivan described the maximal congruences on the semigroup N consisting of all non-negative integers under standard multiplication, and on the semigroup T（X） consisting of all total transformations of an infinite set X under composition. Here, we determine all maximal congruences on the semigroup Zn under multiplication modulo n. And, when Y lohtain in X, we do the same for the semigroup T（X, Y） consisting of all elements of T（X） whose range is contained in Y. We also characterise the minimal congruences on T（X. Y）.
基金Supported by the National Natural Science Foundation of China(11971222)the initial version was posted to arXiv in 2014 with the ID arXiv:1407.8465.
文摘设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1 k Hk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0 −k a a−1 k Hk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0 −k a a−1 k H(2)k 2k+1≡Bp−2(a)(mod p)其中p表示满足a≤r(mod p)的最小非负整数r,Bn(x)与En(x)分别表示次数为n的伯努利多项式与欧拉多项式.
文摘Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.
