Prime Number Problem 5
Model Answer:
For n>1, n! is always an even number. Hence, n!+1 is odd and it cannot have an even divisor.
We prove by the method of contradiction that every prime divisor of n!+1 is an integer greater than n.
Suppose that n!+1 has a prime divisor d less or equal to n. Thus, d divides n!.
But then d divides n! as well as n!+1, and hence, their difference i.e. 1.
A contradiction.
Thus, it is proved that every prime divisor of n!+1 is an odd integer greater than n.
Author
Abhishek Kumar
Abhishek did B. Tech. in Computer Science & Engineering from IIT Kanpur in 2005. Thereafter he joined Indian Revenue Service (IRS) in 2006. He has deep interests in science. He authored the book “Mathematics for Learning Physics”, published by Arihant Publications. This book is designed to be used as the first book after class X and addresses difficulty faced by fresh pass-outs from class X in understanding Physics of class XI on account of non-familiarity with mathematical tools(calculus etc)
Abhishek Kumar
Sep 7th, 2018