Problem 1

Part of 1994 Korea National Olympiad

Problems(2)

integer functions

Source: Korean MO 1994

S

be the set of nonnegative integers. Find all functions

f,g,h: S\rightarrow S

such that f(m\plus{}n)\equal{}g(m)\plus{}h(n), for all

m,n\in S

, and g(1)\equal{}h(1)\equal{}1.

functionalgebra unsolvedalgebra

integer equation

Source: Korean 1994

Consider the equation y^2\minus{}k\equal{}x^3, where

k

is an integer. Prove that the equation cannot have five integer solutions of the form (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4). Also show that if it has the first four of these pairs as solutions, then 63|k\minus{}17.

number theory unsolvednumber theory