There are infinitely many positive integers n {\displaystyle n} such that all of f 1 ( n ) , f 2 ( n ) , … , f k ( n ) {\displaystyle f_{1}(n),f_{2}(n),\ldots ,f_{k}(n)} are simultaneously prime numbers, or There is an integer m > 1 {\displaystyle m>1} (called a fixed divisor) which always divides the product f 1 ( n ) f 2 ( n ) ⋯ f k ( n ) {\displaystyle f_{1}(n)f_{2}(n)\cdots f_{k}(n)} . (Or, equivalently: There exists a prime p {\displaystyle p} such that for every n {\displaystyle n} there is an i {\displaystyle i} such that p {\displaystyle p} divides f i ( n ) {\displaystyle f_{i}(n)} ).