We say that the set of step lengths D⊂Z+={1,2,…} is excellent if it has the following property: If we split the set of integers into two subsets A and Z∖A, at least other set contains element a−d,a,a+d (i.e. {a−d,a,a+d}⊂A or {a−d,a,a+d}∈Z∖A from some integer a∈Z,d∈D.) For example the set of one element {1} is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set {1,2,3,4} is excellent but it has no proper subset which is excellent. modular arithmeticnumber theory unsolvednumber theory