## What is/are Largest Prime?

Largest Prime - • [(3)] For n ∈ N , assume that the largest prime power divisor of n is p a for some a ∈ N.^{[1]}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 $P(1)=1$.

^{[2]}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.

^{[3]}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.

^{[4]}Let $$P_y^+(n)$$Py+(n) denote the largest prime factor p of n with $$p\leqslant y$$p⩽y.

^{[5]}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.

^{[6]}Positive integers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor are studied.

^{[7]}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.

^{[8]}For each n ≥ 1 , let p n be the largest prime less than or equal to n + 1.

^{[9]}

## n − b

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.^{[1]}

## largest prime factor

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 $P(1)=1$.^{[1]}Let $$P_y^+(n)$$Py+(n) denote the largest prime factor p of n with $$p\leqslant y$$p⩽y.

^{[2]}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.

^{[3]}Positive integers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor are studied.

^{[4]}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.

^{[5]}

## largest prime divisor

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.^{[1]}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.

^{[2]}