2

Part of 2022 Switzerland - Final Round

Problems(1)

1^1, 3^3, 5^5, ..., (2n-1)^{2n-1} different remainders divided by 2^n

Source: Switzerland - 2022 Swiss Final Round p2

n

be a positive integer. Prove that the numbers

1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}

all give different remainders when divided by

2^n

number theoryremainderdivides