2021 JMPSC Invitationals Problems/Problem 11
Problem
For some , the arithmetic progression
has exactly
perfect squares. Find the maximum possible value of
Solution
First note that the integers in the given arithmetic progression are precisely the integers which leave a remainder of when divided by
.
Suppose a perfect square is in this arithmetic progression. Observe that the remainders when
,
,
,
, and
are divided by
are
,
,
,
, and
, respectively. Furthermore, for any integer
, [(m+5)^2 = m^2 + 10m + 25 = m^2 + 5(2m + 5),] and so
and
leave the same remainder when divided by
. It follows that the perfect squares in this arithmetic progression are exactly the numbers of the form
and
, respectively.
Finally, the sequence of such squares is [(5\cdot 0 + 2)^2, , (5\cdot 0 + 3)^2, , (5\cdot 1 + 2)^2, , (5\cdot 1 + 3)^2, ,\cdots.] In particular, the first and second such squares are associated with , the third and fourth are associated with
, and so on. It follows that the
such number, which is associated with
, is [(5\cdot 18 + 2)^2 = 92^2 = 9409.] Therefore the arithmetic progression must not reach
. This means the desired answer is
~djmathman