Lucas和Lehmer给出了测定Mersenne数的经典方法[1].在Journal of Number Theory 110(2005)“An elliptic curve test for Mersenne primes”[2]一文中,Benedict又给出了一种对Mersenne数进行素性测的椭圆曲线测试,但并没有给出两种测试...Lucas和Lehmer给出了测定Mersenne数的经典方法[1].在Journal of Number Theory 110(2005)“An elliptic curve test for Mersenne primes”[2]一文中,Benedict又给出了一种对Mersenne数进行素性测的椭圆曲线测试,但并没有给出两种测试运算量的分析与比较.本文根据其原理进行了实现分析,并与经典的Lucas-Lehmer测试进行运算量的比较,结果显示椭圆曲线测试的运算量大于Lucas测试运算量的4倍.展开更多
旨在寻求新梅森素数的大互联网梅森素数搜寻计划GIMPS(Great Internet Mersenne Primes Search)[1]在网格技术的协助下已找到第44个梅森素数。GIMPS是唯一的全球分布计算计划,真正的虚拟组织[2]。梅森素数的计算具有指数复杂性,随着p达...旨在寻求新梅森素数的大互联网梅森素数搜寻计划GIMPS(Great Internet Mersenne Primes Search)[1]在网格技术的协助下已找到第44个梅森素数。GIMPS是唯一的全球分布计算计划,真正的虚拟组织[2]。梅森素数的计算具有指数复杂性,随着p达千万级,所需计算时间须以千、万计算机年计。本文基于梅森素数搜索历程中的原理、技术和算法,探讨网格技术给GIMPS计划带来的突破性进展。展开更多
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
文摘Lucas和Lehmer给出了测定Mersenne数的经典方法[1].在Journal of Number Theory 110(2005)“An elliptic curve test for Mersenne primes”[2]一文中,Benedict又给出了一种对Mersenne数进行素性测的椭圆曲线测试,但并没有给出两种测试运算量的分析与比较.本文根据其原理进行了实现分析,并与经典的Lucas-Lehmer测试进行运算量的比较,结果显示椭圆曲线测试的运算量大于Lucas测试运算量的4倍.
文摘旨在寻求新梅森素数的大互联网梅森素数搜寻计划GIMPS(Great Internet Mersenne Primes Search)[1]在网格技术的协助下已找到第44个梅森素数。GIMPS是唯一的全球分布计算计划,真正的虚拟组织[2]。梅森素数的计算具有指数复杂性,随着p达千万级,所需计算时间须以千、万计算机年计。本文基于梅森素数搜索历程中的原理、技术和算法,探讨网格技术给GIMPS计划带来的突破性进展。
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.
