The twin prime problem revisited

The twin prime problem is defined as the problem of finding a generating function which will predict globally and explicitly the twin prime sequence. The equation of divisibles(or composites)classifies primes and nonprimes by detecting whether the coefficients of divisibility are zeros or nonzeros. One has the options to treat these coefficients as either arithmetic or logical quantities and manipulate them in either sequence algebraic format or set notation format. The prime sequence is the logic complementation of the composite sequence. Since both twin primes and isolated primes are present in the prime sequence, the latter must be filtered off from the prime sequence. This turns out to be a nontrivial mathematical task if only arithmetical operations are used. Up to the present moment, a twin‐prime equation containing only arithmetic operations has not been found. Although primality tests look like arithmetic operations, there are embedded in these algorithms logical decision operators. The twin prime problem becomes tractable only if one permits the use of mixed‐mode mathematics which is defined here as the freedom to switch from arithmetical mode to Boolean mode within the same expression. In order to do this, it is necessary to extend Boolean operations into sequence algebra. The study shows that arithmetic operators lack the sifting and classifying power of Boolean operators. Some problems could benefit from mixed mode mathematics provided one is aware of its limitations.