2007 JBMO Problems/Problem 4
Problem 4
Prove that if is a prime number, then
is not a perfect square.
Solution
By Fermat's Little Theorem, . By quadratic residues, this is true if and only if
, except for
(which doesn't work). Then,
, but this implies
is odd, so
cannot be a perfect square.
See also
2007 JBMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Last Problem | |
1 • 2 • 3 • 4 | ||
All JBMO Problems and Solutions |