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We present new theoretical algorithms that sums the n-ary comparators outputin order to get the permutation indices in order to sort a sequence. Byanalysing the parallel ranking algorithm, we found that the special comparatorsnumber of elements it processes divide the number of elements to be sorted.Using the divide and conquer method, we can express the sorting problem intosumming output of comparators taking a prime number of elements, given thatthis prime number divides the initial disordered sequence length. The number ofsums is directly related to the Maundy cake problem. Furthermore, we provide anew sequence that counts the number of comparators used in the algorithms.

A Curious Link Between Prime Numbers, the Maundy Cake Problem and Parallel Sorting