Investigated the properties of LUCas sequence(LUC), the paper proposed a new variant of (probabilistic) public-key encryption scheme. Security analysis of the proposed encryption schemes shows that its one-wayness is ...Investigated the properties of LUCas sequence(LUC), the paper proposed a new variant of (probabilistic) public-key encryption scheme. Security analysis of the proposed encryption schemes shows that its one-wayness is equivalent to partial LUC discrete logarithm problem in ZN, and for the proposed probabilistic encryption scheme, its semantic security is equivalent to decisional LUC Diffie-Hellman problem in ZN. At last, the efficiency of the proposed schemes is briefly analyzed.展开更多
In this paper, we introduce a new counting function a(m) related to the Lucas number, then use conjecture and induction methods to give an exact formula Ar(N)=α(n), (r=1,2,3) and prove them.
Let A∈N,B∈Z with gcd(A,B)=1,B{-1,0,1}. For the binary recurrence (Lucas sequence) of the form u 0=0, u 1=1, u n+2 =Au n+1 +Bu n, let N 1(A,B,k) be the number of the terms n of |u n|=k, where k∈N. In this paper, usi...Let A∈N,B∈Z with gcd(A,B)=1,B{-1,0,1}. For the binary recurrence (Lucas sequence) of the form u 0=0, u 1=1, u n+2 =Au n+1 +Bu n, let N 1(A,B,k) be the number of the terms n of |u n|=k, where k∈N. In this paper, using a new result of Bilu, Hanrot and Voutier on primitive divisors, we proved that N 1(A,B,k)≤1 except N 1(1,-2,1)=5[n=1,2,3,5,13], N 1(1,-3,1)=3, N 1(1,-5,1)=3,N 1(1,B,1)=2(B{-2,-3,-5}), N 1(12,-55,1)=2, N 1(12,-377,1)=2, N 1(A,B,1)=2(A 2+B=±1, A>1), N 1(1,-2,3)=2, N 1(A,B,A)=2(A 2+2B=±1,A>1. For Lehmer sequence, we got a similar result. In addition, we also obtained some applications of the above results to some Diophantime equations.展开更多
We apply a new, deep theorem of Bilu, Hanrot ＆ Voutier and some fine results on the representation of the solutions of quadratic Diophantine equations to solve completely the exponential Diophantine equation x^2＋（3...We apply a new, deep theorem of Bilu, Hanrot ＆ Voutier and some fine results on the representation of the solutions of quadratic Diophantine equations to solve completely the exponential Diophantine equation x^2＋（3a^2-1）^m = （4a^2-1）^n when 3a^2-1 is a prime or a prime power.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
In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solut...In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）. Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.展开更多
Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of ...Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.展开更多
Sunflower petal is one of the parts of the sunflower which has drawn attention and has several applications these days.These applications justify getting information about physical properties,mechanical properties,dry...Sunflower petal is one of the parts of the sunflower which has drawn attention and has several applications these days.These applications justify getting information about physical properties,mechanical properties,drying trends,etc.in order to design new machines and use new methods to harvest or dry the sunflower petals.For three varieties of sunflower,picking force of petals was measured;number of petals of each head was counted;unit mass and 1000-unit mass of fresh petals were measured and length,width,and projected area of fresh petals were calculated based on image processing technique;frequency distributions of these parameters were modeled using statistical distribution models namely Gamma,Generalized Extreme Value(G.E.V),Lognormal,and Weibull.Results of picking force showed that with increasing number of days after appearing the first petal on each head from 5 to 14 and decreasing loading rate from 150 gmin^-1 to 50 g min^-1 values of picking force were decreased for three varieties,but diameter of sunflower head had different effects on picking force for each variety.Length,width,and number of petals of Dorsefid variety ranged from 38.52 to 95.44 mm,3.80 to 9.28mm and 29 to 89,respectively.The corresponding values ranged from 34.19 to 88.18 mm,4.28 to 10.60 mm and 21 to 89,respectively for Shamshiri variety and ranged from 44.47 to 114.63 mm,7.03 to 20.31 mm and 29 to 89 for Sirena variety.Results of frequency distribution modeling indicated that in most cases,G.E.V and Weibull distributions had better performance than other distributions.展开更多
基金Supported by the 973 State Key Project of China (No.G1999035803)the National Natural Science Foundation of China (No.69931010).
文摘Investigated the properties of LUCas sequence(LUC), the paper proposed a new variant of (probabilistic) public-key encryption scheme. Security analysis of the proposed encryption schemes shows that its one-wayness is equivalent to partial LUC discrete logarithm problem in ZN, and for the proposed probabilistic encryption scheme, its semantic security is equivalent to decisional LUC Diffie-Hellman problem in ZN. At last, the efficiency of the proposed schemes is briefly analyzed.
基金Supported by the Education Department Foundation of Shaanxi Province(03JK213) Supported by the Weinan Teacher's College Foundation(03YKF001)
文摘In this paper, we introduce a new counting function a(m) related to the Lucas number, then use conjecture and induction methods to give an exact formula Ar(N)=α(n), (r=1,2,3) and prove them.
文摘Let A∈N,B∈Z with gcd(A,B)=1,B{-1,0,1}. For the binary recurrence (Lucas sequence) of the form u 0=0, u 1=1, u n+2 =Au n+1 +Bu n, let N 1(A,B,k) be the number of the terms n of |u n|=k, where k∈N. In this paper, using a new result of Bilu, Hanrot and Voutier on primitive divisors, we proved that N 1(A,B,k)≤1 except N 1(1,-2,1)=5[n=1,2,3,5,13], N 1(1,-3,1)=3, N 1(1,-5,1)=3,N 1(1,B,1)=2(B{-2,-3,-5}), N 1(12,-55,1)=2, N 1(12,-377,1)=2, N 1(A,B,1)=2(A 2+B=±1, A>1), N 1(1,-2,3)=2, N 1(A,B,A)=2(A 2+2B=±1,A>1. For Lehmer sequence, we got a similar result. In addition, we also obtained some applications of the above results to some Diophantime equations.
基金the Natural Science Foundation of Guangdong Province (04009801)the Important Science Research Foundation of Foshan University.
文摘We apply a new, deep theorem of Bilu, Hanrot ＆ Voutier and some fine results on the representation of the solutions of quadratic Diophantine equations to solve completely the exponential Diophantine equation x^2＋（3a^2-1）^m = （4a^2-1）^n when 3a^2-1 is a prime or a prime power.
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.
基金NSF of China (No.10571180)the Guangdong Provincial Natural Science Foundation (No.8151027501000114)
文摘In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）. Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.
基金Supported by the Japan Society for the Promotion of Science (JSPS) (No. 14540030) the JSPS Research Fellowships for Young Scientists
文摘Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.
文摘Sunflower petal is one of the parts of the sunflower which has drawn attention and has several applications these days.These applications justify getting information about physical properties,mechanical properties,drying trends,etc.in order to design new machines and use new methods to harvest or dry the sunflower petals.For three varieties of sunflower,picking force of petals was measured;number of petals of each head was counted;unit mass and 1000-unit mass of fresh petals were measured and length,width,and projected area of fresh petals were calculated based on image processing technique;frequency distributions of these parameters were modeled using statistical distribution models namely Gamma,Generalized Extreme Value(G.E.V),Lognormal,and Weibull.Results of picking force showed that with increasing number of days after appearing the first petal on each head from 5 to 14 and decreasing loading rate from 150 gmin^-1 to 50 g min^-1 values of picking force were decreased for three varieties,but diameter of sunflower head had different effects on picking force for each variety.Length,width,and number of petals of Dorsefid variety ranged from 38.52 to 95.44 mm,3.80 to 9.28mm and 29 to 89,respectively.The corresponding values ranged from 34.19 to 88.18 mm,4.28 to 10.60 mm and 21 to 89,respectively for Shamshiri variety and ranged from 44.47 to 114.63 mm,7.03 to 20.31 mm and 29 to 89 for Sirena variety.Results of frequency distribution modeling indicated that in most cases,G.E.V and Weibull distributions had better performance than other distributions.
