Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for <i>π</i>

Abstract

In our earlier publication we have shown how to compute by iteration a rational number u2,k in the two-term Machin-like formula for π of the kind π4=2k−1arctan1u1,k+arctan1u2,k,k∈Z,k≥1, where u1,k can be chosen as an integer u1,k=ak/2−ak−1 with nested radicals defined as ak=2+ak−1 and a0=0. In this work, we report an alternative method for determination of the integer u1,k. This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set ak in computation of the integer u1,k. Mathematica programs validating these results are presented.

Keywords

• constant π
• Machin-like formula
• Lehmer’s measure
• surd number
• Ramanujan’s nested radical