黎曼猜想和素数分布有什么深层联系？
What deep link ties the Riemann hypothesis to prime distribution?
黎曼ζ函数的零点位置决定了 π(x) 与 x/ln x 之间误差的大小；若猜想成立，素数分布会异常规则。它是数学中最著名的未解问题之一。
The zeros of the Riemann zeta function control the error between π(x) and x/ln x; if the hypothesis holds, prime distribution is exceptionally regular. It is among math's most famous unsolved problems.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#素数 #黎曼
素数定理告诉我们素数大约有多密？
What does the prime number theorem say about how dense primes are?
它说小于 x 的素数个数 π(x) 渐近于 x/ln x，即素数在自然数中越来越稀，但永不消失。大数附近相邻素数平均间隔约 ln x。
It says the count of primes below x, π(x), is asymptotic to x/ln x—primes thin out but never vanish. Near large x the average gap between primes is about ln x.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#素数 #分布
哥德巴赫猜想说的是什么？
What does Goldbach's conjecture say?
它猜想：每个大于2的偶数都可写成两个素数之和，如 8=3+5。已被计算机验证到极大范围，但仍无一般证明。
It conjectures every even number greater than 2 is the sum of two primes, e.g. 8=3+5. Verified by computers to enormous ranges, yet still unproven generally.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#素数 #猜想
什么是完全数？它和素数有什么关系？
What is a perfect number, and its link to primes?
完全数是等于自身所有真因子之和的数，如 6=1+2+3、28=1+2+4+7+14。欧几里得发现：当 2^p−1 是素数（梅森素数）时，2^(p−1)(2^p−1) 是完全数。
A perfect number equals the sum of its proper divisors, e.g. 6=1+2+3, 28=1+2+4+7+14. Euclid showed 2^(p−1)(2^p−1) is perfect whenever 2^p−1 is prime (a Mersenne prime).
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#完全数 #梅森
怎么用欧几里得的方法证明素数有无穷多个？
How does Euclid's argument prove there are infinitely many primes?
假设素数只有有限个，把它们全乘起来再加1，得到的新数不能被其中任何一个整除，于是要么是新素数、要么含新素因子，矛盾。所以素数无穷。
Suppose finitely many primes exist; multiply them all and add 1. The result is not divisible by any of them, so it is either a new prime or has a new prime factor—contradiction. Hence infinitely many primes.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#素数 #证明