MathDB

4

Part of 2019 New Zealand MO

Problems(2)

122^n - 102^n - 21^n is one less than a multiple of 2020

Source: New Zealand MO 2019 Round 1 p4

9/20/2021
Show that the number 122n−102n−21n122^n - 102^n - 21^n is always one less than a multiple of 20202020, for any positive integer nn.
number theorydividesdivisiblemultiple
n2^k -7 is a perfect square

Source: 2019 New Zealand MO Round 2 p4 NZMO

9/22/2021
Show that for all positive integers kk, there exists a positive integer n such that n2k−7n2^k -7 is a perfect square.
number theoryPerfect Square