MathDB

4

Part of 2010 IMC

Problems(2)

IMC 2010 - Problem 4

Source:

7/26/2010
Let a,ba,b be two integers and suppose that nn is a positive integer for which the set Z\{axn+byn∣x,y∈Z}\mathbb{Z} \backslash \{ax^n + by^n \mid x,y \in \mathbb{Z}\} is finite. Prove that n=1n=1.
functionabstract algebraquadraticsset theorynumber theory unsolvednumber theory
IMC 2010, Problem 4, Day 2

Source:

7/27/2010
Let AA be a symmetric m×mm\times m matrix over the two-element field all of whose diagonal entries are zero. Prove that for every positive integer nn each column of the matrix AnA^n has a zero entry.
linear algebramatrixgraph theorylinear algebra unsolved