An iteration procedure for a two-term Machin-like formula for pi with small Lehmer's measure

Abstract

In this paper we present a two-term Machin-like formula for pi \[\fracπ{4} = 2^{k - 1}\arctan\left(\frac{1}{u_1}\right) + \arctan\left(\frac{1}{u_2}\right)\] with small Lehmer's measure $e\approx 0.245319$ and describe iteration procedure for simplified determination of the required rational number $u_2$ at $k=27$ and $u_1=85445659$. With these results we obtained a formula that has no irrational numbers involved in computation and provides $16$ digits of pi at each increment by one of the summation terms. This is the smallest Lehmer's measure ever reported for the Machin-like formulas for pi.

20 pages

Cited by in corpus (1)

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