A Partition Problem

AbstractFor positive integers l, n, k we say that M = M(n, k) = {n, n + 1, ..., n + k} has an l-partition, if there is A ⊂ M(n, k) with l∑a ∈ Aa = ∑m ∈ Mm. Moreover, define B(l) = {M(n, k) : M has an l-partition}, and K(l) = minM ∈ B(l) (|M| − 1). We show that M(n, k) ∈ B(2) iff k ≡ 3 mod 4 or 2 | k, n ≡ k/2 mod 2, 4n ≤ k2. Then we prove an explicit formula for K(pd), where p is prime; finally, we introduce a method of determining K(r) for arbitrary r ∈ N, particularly for r = pq with primes p and q