A Circular Binary Search

In a standard binary search, the binary representation of the index of an element in an ordered linear array is recovered serially bit by bit. For an array of N elements, the index of an element is recovered, in principle, by assigning to each element one value out of log₂N possibilities. It is shown in this paper that by arranging 2ⁿ—1 elements in a circular array, the bits of the binary representation of the index of an element are all recovered simultaneously based on assigning to each element one value out of two possibilities. The main theoretical result is showing that the parity of an integer X is trivially recovered from of the parity of the Hamming weight of the binary representation of X, X+ 1, X+ 2, and X+ 3, whereas, on the other hand, the parity of the Hamming weight of the binary representation of an integer is consistent with modular arithmetic considerations.