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24 June 2024 

   

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Three essays on Machin's type formulas  Armengol Gasull
; Florian Luca
; Juan L. Varona
;  Date: 
1 Feb 2023  Abstract:  We study three questions related to Machin’s type formulas. The first one
gives all two terms Machin formulas where both arctangent functions are
evaluated $2$integers, that is values of the form $b/2^a$ for some integers
$a$ and~$b$. These formulas are computationally useful because multiplication
or division by a power of two is a very fast operation for most computers. The
second one presents a method for finding infinitely many formulas with $N$
terms. In the particular case $N=2$ the method is quite useful. It recovers
most known formulas, gives some new ones, and allows to prove in an easy way
that there are two terms Machin formulas with Lehmer measure as small as
desired. Finally, we correct an oversight from previous result and give all
Machin’s type formulas with two terms involving arctangents of powers of the
golden section.  Source:  arXiv, 2302.00154  Services:  Forum  Review  PDF  Favorites 


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