Easy Proofs of Some Borwein Algorithms for π.

The article reports on the proof of some Borwein algorithms for π. It states that Jonathan and Peter Borwein derived many efficient algorithms for computing π in 1987. However, their proofs are difficult for the nonspecialist in the theory of elliptic modular functions. This can be seen by using a formula of Gauss's and elementary algebra which is able to prove the correctness of the two of them. There are two algorithms that was considered: quadratic algorithm and quartic algorithm.