A problem about divisibility

Show that there exists infinitely many positive integers such that .

I am begging for seeing all satisfying the above requirement . The job should belong to computer but i have no idea how to design a computer programme to look for the numbers .

Here's my attempt to the proof :

Let be the sequence with . Then i find that . Moreover , interestingly i can factorize into . But and is prime to . Therefore , if , we also have then it finishes the proof by induction .

Since the sequence contains the 'very large' integers ( !) , i wonder if it is true that the numbers could hardly be found in the integer line .... But i think that is not true , i believe by solving quadratic congruences we can construct many such integers !

Thank you