Problem 7 Let k M1 Mk1 be positive integers each greater

Problem 7. Let k, M1, . . . ,Mk+1 be positive integers, each greater than 1. Assume that the integers M1, . . . , _{}^{}Mk+1 are pairwise relatively prime. Prove that there exist k+1 consecutive integers n,n+1,...,n+k such that M^{2}i+1 | (n+i) for each 0ik.

Solution

here m1,m2-----mk+1 are positive integers.

and each is grater than 1 means k,m1----mk+1>1

k = 1, this is just exactly the definition of the function. Now lets assume that the equation holds for some k, and try to prove it for k + 1.

The number of integers 1 n m(k + 1)

here k=k1,k2,k3-----k+1

there are integers that must be grater than 1.

n,n+1,-----n+k are also integers and prime numbers.