Some Related Functions to Integer GCD and Coprimality

We generalize a formula of B. Litow [Parallel Complexity of Integer Coprimality, in Electronic Colloquium on Computational Complexity, Report No. 9, 1998.] and propose several new formula linked with the parallel Integer Coprimality, Integer GCD and Modular Inverse problems as well. Particularly, we find a new trigonometrical definition of the GCD of two integers a , b ⩾ 1 :(1) gcd ( a , b ) = 1 π ∫ 0 π cos [ ( b − a ) x ] sin 2 ( a b x ) sin ( a x ) sin ( b x ) d x . We also suggest a generalization of the GCD function to real numbers.