2008 iTest Problems/Problem 94
Problem
Find the largest prime number less than that is a divisor of some integer in the infinite
sequence
Solution
The largest prime number less than is
; we claim that this is the answer. Indeed, we claim that the
th term divides
, where
is prime (and hence relatively prime to
).
To do so, we claim that
holds, and since is prime the result follows. Indeed,
, where
denotes the fractional part of a number. So
becomes
By Fermat's Little Theorem, we have , so
. Also,
is equivalent to the remainder when
is divided by
, and by Fermat's Little Theorem again, we have
. Hence, equation
reduces to
as desired.