Rectangular Heffter arrays: a reduction theorem

Let m,n,s,k be four integers such that 3?s?n, 3?k?m and ms=nk. Set d=gcd?(s,k). In this paper we show how one can construct a Heffter array H(m,n;s,k) starting from square H(nk/d;d) whose elements belong to d consecutive diagonals. As an example of application method, prove there exists integer in each the following cases: (i) d?0(mod4); (ii) 5?d?1(mod4) nk?3(mod4); (iii) d?2(mod4) nk?0(mod4); (iv) d?3(mod4) nk?0,3(mod4). The same method applied also for signed magic arrays SMA(m,n;s,k) rectangles MR(m,n;s,k). fact, when d?2, MR(m,n;s,k) either d?2 is even or d?3 nk are odd. We provide constructions k odd s?0(mod4).