The Representation of Some Integers as a Subset Sum

Let A ⊆ N . The cardinality (the sum of the elements) of A will be denoted by | A | (Σ( A )). Let m ∈ N and p be a prime. Let A ⊆ {1, 2,…, p }. We prove the following results. If | A | ⩾ [( p + m −2)/ m ]+ m , then for every integer x such that 0 ⩽ x ⩽ p − 1, there is B ⊆ A such that | B | = m and Σ( B ) ≡ x mod p . Moreover, the bound is attained. If | A | ⩾ [( p + m −2)/ m ]+ m !, then there is B ⊆ A such that | B | ≡ 0 mod m and Σ( B ) = ( m − 1)! p . If | A | ⩾ [( p + 1)/3]+29, then for every even integer x such that 4 p ⩽s x ⩽ p ( p + 170)/48, there is S ⊆ A such that x = Σ( S ). In particular, for every even integer a ⩾ 2 such that p ⩾ 192 a − 170, there are an integer j ⩾ 0 and S ⊆ A such that Σ( S ) = a j +1 .