Title:

Problems related to number theory : sumanddistance systems, reversible square matrices and divisor functions

We say that two sets $A$ and $B$, each of cardinality $m$, form an $m+m$ \emph{sumanddistance system} $\{A,B\}$ if the sumanddistance set $A^*B$ comprised of all the absolute values of the sums and distances $a_i\pm b_j$ contains either the consecutive odd integers $\{1,3,5,\ldots 4m^21\}$ or with the inclusion of the set elements themselves, the consecutive integers $\{1,2,3,\ldots,2m(m+1)\}$ (an inclusive sumanddistance system). Sumanddistance systems can be thought of as a discrete analogue of the union of a Minkowski sum system with a Minkowski difference system. We show that they occur naturally within a traditional reversible square matrix, where conjugation with a specific orthogonal symmetric involution, always reveals a sumanddistance system within the block structure of the conjugated matrix. Moreover, we show that the block representation is an algebra isomorphism. Building upon results of Ollerenshaw, and Br\'ee, for a fixed dimension $n$, we establish a bijection between the set of sumanddistance systems and the set of traditional principal reversible square matrices of size $n\times n$. Using the $j$th nontrivial divisor function $c_j (n)$, which counts the total number of proper ordered factorisations of the integer $n= p_1^{a_1}\ldots p_t^{a_t}$ into $j$ parts, we prove that the total number of $n+n$ principal reversible square matrices, and so sumanddistance systems, $N_n$, is given by \[ N_n = \sum_{j=1}^{\Omega(n)} \left( c_j(n)^2 +c_{j+1}(n)c_j(n) \right)=\sum_{j=1}^{\Omega(n)} c_j^{(0)}(n) c_j^{(1)}(n). \] \[=\sum_{j=1}^{\Omega(n)} \left(\sum^j_{i=1}(1)^{ji}{j \choose i} \prod_{k=1}^t {a_k +i 1 \choose i1}\right ) \left ( \sum^j_{i=0}(1)^{ji}{j \choose i} \prod_{k=1}^t {a_k +i \choose i}\right), \] where $\Omega(n)=a_1 + a_2 + \ldots + a_t$ is the total number of prime factors (including repeats) of $n$. Further relations between the divisor functions and their Dirichlet series are deduced, as well as a construction algorithm for all sumanddistance systems of either type. Superalgebra structures relating to the matrix symmetry properties are identified, including those for the reversible and mostperfect square matrices of those considered by Ollerenshaw and Br\'ee. For certain symmetry types, links between the block representation constructed from a sumanddistance system, and quadratic forms are also established.
