Let G be a graph, and g and f be integer valued functions defined on V(G) which satisfy g(x)≤f(x) and g(x)≡f(x)(mod 2) for all x∈V(G). Then a spanning subgraph F of G is called a {g,g+2,…,f} -factor if deg_F(x)∈{...Let G be a graph, and g and f be integer valued functions defined on V(G) which satisfy g(x)≤f(x) and g(x)≡f(x)(mod 2) for all x∈V(G). Then a spanning subgraph F of G is called a {g,g+2,…,f} -factor if deg_F(x)∈{g(x),g(x)+2,…,f(x)} for all x∈V(G), when g(x)=1 for all x∈V(G), such a factor is called (1,f) -odd-factor. We give necessary and sufficient conditions for a graph G to have a {g,g+2,…,f} -factor and a (1,f) -odd-factor which contains an arbitrarily given edge of G, from that we derive some interesting results.展开更多
For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x, y) of the equation y^2 = px(x^2 + 2). In this paper, using some properties of binary quartic Diophantine equations, we pro...For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x, y) of the equation y^2 = px(x^2 + 2). In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7(mod 8), then N(p) = 0; ifp ≡ 1(mod 8), then N(p) 〈 1; if p〉 3 andp ≡ 3(rood 8), then N(p) ≤ 2.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
In this paper, the possible value of the differential uniformity of a function over finite fields is discussed. It is proved that, the differential uniformity of a function over Fq can be any even integer between 2 an...In this paper, the possible value of the differential uniformity of a function over finite fields is discussed. It is proved that, the differential uniformity of a function over Fq can be any even integer between 2 and q when q is even; and it can be any integer between 1 and q except q-1 when q is odd. Moreover, for any possible differential uniformity t, an explicit construction of a differentially t-uniform function is given.展开更多
Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that(k+l1)2,(k+l2)2,…,(k+lg)2 can not be expressed as 2n+...Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that(k+l1)2,(k+l2)2,…,(k+lg)2 can not be expressed as 2n+pαis at least c(g)x,where p is an odd prime and the constant c(g)depends only on g.展开更多
The Hilbert genus field of the real biquadratic field K=Q(√δ,√d)is described by Yue(2010)and Bae and Yue(2011)explicitly in the case&=2 or p with p=1 mod 4 a prime and d a squarefree positive integer.In this...The Hilbert genus field of the real biquadratic field K=Q(√δ,√d)is described by Yue(2010)and Bae and Yue(2011)explicitly in the case&=2 or p with p=1 mod 4 a prime and d a squarefree positive integer.In this article,we describe explicitly the Hilbert genus field of the imaginary biquadratic field K=Q(√δ,√d),whereδ=-1,-2 or-p with p=3 mod 4 a prime and d any squarefree integer.This completes the explicit construction of the Hilbert genus field of any biquadratic field which contains an imaginary quadratic subfield of odd class number.展开更多
Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,...Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q.展开更多
文摘Let G be a graph, and g and f be integer valued functions defined on V(G) which satisfy g(x)≤f(x) and g(x)≡f(x)(mod 2) for all x∈V(G). Then a spanning subgraph F of G is called a {g,g+2,…,f} -factor if deg_F(x)∈{g(x),g(x)+2,…,f(x)} for all x∈V(G), when g(x)=1 for all x∈V(G), such a factor is called (1,f) -odd-factor. We give necessary and sufficient conditions for a graph G to have a {g,g+2,…,f} -factor and a (1,f) -odd-factor which contains an arbitrarily given edge of G, from that we derive some interesting results.
基金Foundation item: Supported by the Natural Science Foundation of Shaanxi Province(2009JM1006)
文摘For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x, y) of the equation y^2 = px(x^2 + 2). In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7(mod 8), then N(p) = 0; ifp ≡ 1(mod 8), then N(p) 〈 1; if p〉 3 andp ≡ 3(rood 8), then N(p) ≤ 2.
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.
基金supported by National Natural Science Foundation of China(Grant Nos.61070215 and 61272484)the National Basic Research Program of China(Grant No.2013CB338002)the open research fund of Science and Technology on Information Assurance Laboratory(Grant No.KJ-12-02)
文摘In this paper, the possible value of the differential uniformity of a function over finite fields is discussed. It is proved that, the differential uniformity of a function over Fq can be any even integer between 2 and q when q is even; and it can be any integer between 1 and q except q-1 when q is odd. Moreover, for any possible differential uniformity t, an explicit construction of a differentially t-uniform function is given.
基金supported by the National Natural Science Foundation of China(Nos.10771103,10801075)the Natural Science Foundation of Huaihai Institute of Technology(No.KQ10002)
文摘Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that(k+l1)2,(k+l2)2,…,(k+lg)2 can not be expressed as 2n+pαis at least c(g)x,where p is an odd prime and the constant c(g)depends only on g.
基金supported by National Key Basic Research Program of China(Grant No.2013CB834202)National Natural Science Foundation of China(Grant No.11171317)
文摘The Hilbert genus field of the real biquadratic field K=Q(√δ,√d)is described by Yue(2010)and Bae and Yue(2011)explicitly in the case&=2 or p with p=1 mod 4 a prime and d a squarefree positive integer.In this article,we describe explicitly the Hilbert genus field of the imaginary biquadratic field K=Q(√δ,√d),whereδ=-1,-2 or-p with p=3 mod 4 a prime and d any squarefree integer.This completes the explicit construction of the Hilbert genus field of any biquadratic field which contains an imaginary quadratic subfield of odd class number.
基金supported by National Natural Science Foundation of China (Grant No. 11271212)
文摘Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q.