MathDB

3

Part of 2009 Korea National Olympiad

Problems(2)

Infinite descent

Source: Korea National Olympiad 2009 Problem 3

9/9/2012
Let nn be a positive integer. Suppose that the diophantine equation zn=8x2009+23y2009z^n = 8 x^{2009} + 23 y^{2009} uniquely has an integer solution (x,y,z)=(0,0,0)(x,y,z)=(0,0,0). Find the possible minimum value of nn.
number theoryDiophantine equationnumber theory proposed
2^n-1 does not divide 3^n-1

Source: Korea National Olympiad 2009 Problem 7

9/9/2012
For all positive integer n≥2 n \ge 2 , prove that 2n−1 2^n -1 can't be a divisor of 3n−1 3^n -1 .
number theory proposednumber theory