Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos^2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the seque...Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos^2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the sequence has a primitive divisor in Z[2 cos(2π/7)].As a consequence we deduce that for any sufficiently large n there exists a prime power q such that the groupcan be generated by a pair x,y with χ~2=y^3=(xy)~7=1 and the order of the commutator[x,y]is exactly n.The latter result answers in affirmative a question of Holt and Plesken.展开更多
A conjecture posed by Melham for Un2k+Un+12k in [2] is considered in this article.Several positive results on Vn+s2k +(-1)k(p2-4)kUn2k and V n2k+(-1)k(p2-4)kUn+s2k are achieved which generalize existed results on V n+...A conjecture posed by Melham for Un2k+Un+12k in [2] is considered in this article.Several positive results on Vn+s2k +(-1)k(p2-4)kUn2k and V n2k+(-1)k(p2-4)kUn+s2k are achieved which generalize existed results on V n+12-(p2-4)Un2 and Vn2-(p2-4)Un+12.展开更多
The Diophantine equation X（ X ＋ 1 ） （ X ＋ 2 ） （ X ＋ 3 ） = 14Y（ Y ＋ 1 ） （ Y ＋ 2 ） （ Y ＋ 3 ） still remains open. Using recurrence sequence, Maple software, Pell equation and quadraric residue, this pap...The Diophantine equation X（ X ＋ 1 ） （ X ＋ 2 ） （ X ＋ 3 ） = 14Y（ Y ＋ 1 ） （ Y ＋ 2 ） （ Y ＋ 3 ） still remains open. Using recurrence sequence, Maple software, Pell equation and quadraric residue, this paper proved it has only two positive integer solutions, i. e., （X,Y） = （5,2） ,（7,3）.展开更多
Hepatocellular carcinoma（HCC）is one of the leading causes of cancer related death in Asia and Africa（1）.It reflects the high burden of hepatitis B virus（HBV）infection in these areas.Curative treatments of HCC as...Hepatocellular carcinoma（HCC）is one of the leading causes of cancer related death in Asia and Africa（1）.It reflects the high burden of hepatitis B virus（HBV）infection in these areas.Curative treatments of HCC as radiofrequency ablation and resection are impaired by a high rate of tumor recurrence.However,most of the time,HCC is frequently diagnosed at advanced stages where only展开更多
Suppose thatx=|x(n)|n∈? is a sequence of real numbers. For eachp∈?,x p =|x p (n)|n∈?is the resulting sequence ofx throughp times median filterings with window 2k+1. It is proved that whenp→∞, bothx (2p) andx(2 p}...Suppose thatx=|x(n)|n∈? is a sequence of real numbers. For eachp∈?,x p =|x p (n)|n∈?is the resulting sequence ofx throughp times median filterings with window 2k+1. It is proved that whenp→∞, bothx (2p) andx(2 p}-1) are convergent. Thus the problem of convergence of the median filters of infinite-length sequences is completely solved.展开更多
Properties of third-order recurrence sequences were investigated and a new variant of the GH public-key cryptosystem,which was further improved to be a probabil-istic public-key cryptosystem,was proposed.Then security...Properties of third-order recurrence sequences were investigated and a new variant of the GH public-key cryptosystem,which was further improved to be a probabil-istic public-key cryptosystem,was proposed.Then security analysis of the proposed scheme was provided and it was proved that the one-wayness of the proposed scheme is equivalent to partial discrete logarithm and its semantic se-curity is equivalent to decisional Diffie-Hellman problem in ring extension.Finally,efficiency analysis of the proposed scheme was provided,and that these two encryption schemes need to transfer 2log N and 4log N bits data re-spectively.展开更多
基金supported by the Government of the Russian Federation (Grant No. 14.Z50.31.0030)
文摘Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos^2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the sequence has a primitive divisor in Z[2 cos(2π/7)].As a consequence we deduce that for any sufficiently large n there exists a prime power q such that the groupcan be generated by a pair x,y with χ~2=y^3=(xy)~7=1 and the order of the commutator[x,y]is exactly n.The latter result answers in affirmative a question of Holt and Plesken.
基金Foundation item： Supported by the National Natural Science Foundation of China（10771093）
文摘A conjecture posed by Melham for Un2k+Un+12k in [2] is considered in this article.Several positive results on Vn+s2k +(-1)k(p2-4)kUn2k and V n2k+(-1)k(p2-4)kUn+s2k are achieved which generalize existed results on V n+12-(p2-4)Un2 and Vn2-(p2-4)Un+12.
基金The Natural Science Foundation of Chongqing University of Post and Telecommunications (No.A2008-40)
文摘The Diophantine equation X（ X ＋ 1 ） （ X ＋ 2 ） （ X ＋ 3 ） = 14Y（ Y ＋ 1 ） （ Y ＋ 2 ） （ Y ＋ 3 ） still remains open. Using recurrence sequence, Maple software, Pell equation and quadraric residue, this paper proved it has only two positive integer solutions, i. e., （X,Y） = （5,2） ,（7,3）.
基金supported by HECAM (BPI), EBCI, INCa (Wnt HCC project). J-C.N.supported by a fellowship from INCa
文摘Hepatocellular carcinoma（HCC）is one of the leading causes of cancer related death in Asia and Africa（1）.It reflects the high burden of hepatitis B virus（HBV）infection in these areas.Curative treatments of HCC as radiofrequency ablation and resection are impaired by a high rate of tumor recurrence.However,most of the time,HCC is frequently diagnosed at advanced stages where only
基金Project supported by the National Natural Science Foundation of China (Grant No. 16971047)
文摘Suppose thatx=|x(n)|n∈? is a sequence of real numbers. For eachp∈?,x p =|x p (n)|n∈?is the resulting sequence ofx throughp times median filterings with window 2k+1. It is proved that whenp→∞, bothx (2p) andx(2 p}-1) are convergent. Thus the problem of convergence of the median filters of infinite-length sequences is completely solved.
基金supported by the National Natural Science Foundation of China(No.90412011)the Hi-Tech Research and Development Program of China(No.2002AA143021)。
文摘Properties of third-order recurrence sequences were investigated and a new variant of the GH public-key cryptosystem,which was further improved to be a probabil-istic public-key cryptosystem,was proposed.Then security analysis of the proposed scheme was provided and it was proved that the one-wayness of the proposed scheme is equivalent to partial discrete logarithm and its semantic se-curity is equivalent to decisional Diffie-Hellman problem in ring extension.Finally,efficiency analysis of the proposed scheme was provided,and that these two encryption schemes need to transfer 2log N and 4log N bits data re-spectively.
