## Bang (1886), Zsigmondy (1892) and Birkhoff and Vandiver (1904) initiated the study of the largest prime divisors of sequences of the form a n − b n , denoted P ( a n − b n ) , by essentially proving that for integers a > b > 0 , P ( a n − b n ) ≥ n + 1 for every n > 2. Bang (1886)、Zsigmondy (1892) 和 Birkhoff 和 Vandiver (1904) 通過本質上證明對於整數 a > b > 0 , P ( a n − b n ) ≥ n + 1 對於每個 n > 2。

Prime divisors of sparse values of cyclotomic polynomials and Wieferich primes

## Denote by \$\mathbb{P}\$ the set of all prime numbers and by \$P(n)\$ the largest prime factor of positive integer \$n\geq 1\$ with the convention \$P(1)=1\$. 用 \$\mathbb{P}\$ 表示 所有素數的集合和由 \$P(n)\$ 正整數\$n\geq 1\$的最大素數 與約定 \$P(1)=1\$。

ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES

## Let \$\$P_y^+(n)\$\$Py+(n) denote the largest prime factor p of n with \$\$p\leqslant y\$\$p⩽y. 令 \$\$P_y^+(n)\$\$Py+(n) 用 \$\$p\leqslant y\$\$p⩽y 表示 n 的最大素因子 p。

Sur les plus grands facteurs premiers inférieur à y d’entiers consécutifs

## Results about the prime factorization of \$g(n)\$ allow a reduction of the upper bound on the largest prime divisor of \$g(n)\$ to \$1. 關於 \$g(n)\$ 的素數分解的結果允許將 \$g(n)\$ 的最大素數除數的上限減少到 \$1。

The Largest Prime Dividing the Maximal Order of an Element of \$S_n\$.

## Bang (1886), Zsigmondy (1892) and Birkhoff and Vandiver (1904) initiated the study of the largest prime divisors of sequences of the form a n − b n , denoted P ( a n − b n ) , by essentially proving that for integers a > b > 0 , P ( a n − b n ) ≥ n + 1 for every n > 2. Bang (1886)、Zsigmondy (1892) 和 Birkhoff 和 Vandiver (1904) 通過本質上證明對於整數 a > b > 0 , P ( a n − b n ) ≥ n + 1 對於每個 n > 2。

Prime divisors of sparse values of cyclotomic polynomials and Wieferich primes

## • [(3)] For n ∈ N , assume that the largest prime power divisor of n is p a for some a ∈ N. • [(3)] 對於 n ∈ N，假設對於一些 a ∈ N，n 的最大素數冪因數是 p a。

On generalized Erdős-Ginzburg-Ziv constants of Cnr

## Denote by \$\mathbb{P}\$ the set of all prime numbers and by \$P(n)\$ the largest prime factor of positive integer \$n\geq 1\$ with the convention \$P(1)=1\$. 用 \$\mathbb{P}\$ 表示 所有素數的集合和由 \$P(n)\$ 正整數\$n\geq 1\$的最大素數 與約定 \$P(1)=1\$。

ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES

10.1090/S0025-5718-1995-1270619-3

## Results about the prime factorization of \$g(n)\$ allow a reduction of the upper bound on the largest prime divisor of \$g(n)\$ to \$1. 關於 \$g(n)\$ 的素數分解的結果允許將 \$g(n)\$ 的最大素數除數的上限減少到 \$1。

The Largest Prime Dividing the Maximal Order of an Element of \$S_n\$.

## Bang (1886), Zsigmondy (1892) and Birkhoff and Vandiver (1904) initiated the study of the largest prime divisors of sequences of the form a n − b n , denoted P ( a n − b n ) , by essentially proving that for integers a > b > 0 , P ( a n − b n ) ≥ n + 1 for every n > 2. Bang (1886)、Zsigmondy (1892) 和 Birkhoff 和 Vandiver (1904) 通過本質上證明對於整數 a > b > 0 , P ( a n − b n ) ≥ n + 1 對於每個 n > 2。

Prime divisors of sparse values of cyclotomic polynomials and Wieferich primes

## Let \$\$P_y^+(n)\$\$Py+(n) denote the largest prime factor p of n with \$\$p\leqslant y\$\$p⩽y. 令 \$\$P_y^+(n)\$\$Py+(n) 用 \$\$p\leqslant y\$\$p⩽y 表示 n 的最大素因子 p。

Sur les plus grands facteurs premiers inférieur à y d’entiers consécutifs

## We apply our results firstly to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors \$P^{+}(n)\$, \$P^{+}(n+1), P^{+}(n+2)\$ of three consecutive integers. 我們首先應用我們的結果以更強有力的形式回答 Erdős 和 Pomerance 關於最大素因子 \$P^{+}(n)\$、\$P^{+}(n+1)、P 的相對排序的問題^{+}(n+2)\$ 的三個連續整數。

Value patterns of multiplicative functions and related sequences

## Positive integers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor are studied. 研究等於從最小素數到最大素數的連續素數的冪的和的正整數。

Straddled numbers: numbers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor

## Denote by ℙ the set of all primes and by P(n) the largest prime factor of integer n ≥ 1 with the convention P(1) = 1. 用 ℙ 表示所有素數的集合，用 P(n) 表示整數 n ≥ 1 的最大素因數，約定 P(1) = 1。

On values taken by the largest prime factor of shifted primes (II)

## For each n ≥ 1 , let p n be the largest prime less than or equal to n + 1. 對於每個 n ≥ 1 ，令 p n 是小於或等於 n + 1 的最大素數。

Equality of numerical ranges of matrix powers