摘要
我们给出了关于模素数原根的许多新观察。对奇素数p与整数c,我们建立了一个关于∑g(g+c/p)的定理,其中g跑遍1,…,p-1中模p的原根,(·/p)为Legendre符号。在我们数值计算的基础上,我们形成了35个关于模素数原根的猜想。例如:我们猜测对任何素数p有个模p的原根g<p使得g-1为平方数,还猜测对素数p>3有素数g<p使得Bernoulli数Bq-1为模p的原根。我们也有与模素数的平方非剩余以及一些组合序列的本原素因子有关的观察。例如:在启发式论据基础上,我们猜测对素数p> 3有个Fibonacci数Fk<p/2为模p的平方非剩余,这蕴含着有多项式时间算法可对模素数p> 3的平方剩余找出其模p的平方根。
We make many new observations on primitive roots modulo primes.For an odd prime p and an integer c,we establish a theorem concerning ∑g(g+c/p) ,where g runs over all the primitive roots modulo p among 1,…,p-1,and (·/p) denotes the Legendre symbol.On the basis of our numerical computations,we formulate 35 conjectures involving primitive roots modulo primes.For example,we conjecture that for any prime p there is a primitive root g <p modulo p with g-1 a square,and that for any prime p> 3 there is a prime q <p with the Bernoulli number Bq-1 a primitive root modulo p.We also make related observations on quadratic nonresidues modulo primes and primitive prime divisors of some combinatorial sequences.For example,based on heuristic arguments we conjecture that for any prime p> 3 there exists a Fibonacci number Fk <p/2 which is a quadratic nonresidue modulo p;this implies that there is a deterministic polynomial time algorithm to find square roots of quadratic residues modulo a prime p> 3.
作者
孙智伟
Sun Zhi-Wei(Department of Mathematics,Nanjing University,Nanjing 210093)
出处
《南京大学学报（数学半年刊）》
2019年第2期108-133,共26页
Journal of Nanjing University(Mathematical Biquarterly)
基金
Supported by the National Natural Science Foundation(grant 11571162)of China.
关键词
模素数的原根
有限域
平方剩余
组合序列
本原素因子
Primitive root modulo a prime
finite field
quadratic residue
combinatorial sequence
primitive prime divisor
