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 log₂ N possibilities. It is shown here 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 n assigning to each element one value out of two possibilities. The main theoretical result shows that the parity of an integer X is trivially recovered from 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