Editing Stoneham number

In [[mathematics]], the '''Stoneham numbers''' are a certain class of [[real number]]s, named after [[mathematician]] [[Richard G. Stoneham]] (1920–1996).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Stoneham Number |url=https://mathworld.wolfram.com/StonehamNumber.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}</ref> For [[coprime]] numbers ''b'', ''c'' &gt; 1, the Stoneham number &alpha;<sub>''b'',''c''</sub> is defined as

:<math>\alpha_{b,c} = \sum_{n=c^k>1} \frac{1}{b^nn} = \sum_{k=1}^\infty \frac{1}{b^{c^k}c^k}</math>

It was shown by Stoneham in 1973 that &alpha;<sub>''b'',''c''</sub> is ''b''-[[normal number|normal]] whenever ''c'' is an odd [[prime number|prime]] and ''b'' is a [[primitive root modulo n|primitive root]] of ''c''<sup>2</sup>. In 2002, Bailey & Crandall showed that coprimality of ''b'', ''c'' > 1 is sufficient for ''b''-normality of &alpha;<sub>''b'',''c''</sub>.<ref>{{Cite journal |last1=Bailey |first1=David H. |last2=Crandall |first2=Richard E. |date=2002 |title=Random Generators and Normal Numbers |url=https://www.tandfonline.com/doi/abs/10.1080/10586458.2002.10504704 |journal=Experimental Mathematics |volume=11|issue=4 |pages=527–546 |doi=10.1080/10586458.2002.10504704 |s2cid=8944421 }}</ref>

== References ==
{{reflist}}
*{{citation
 | last1 = Bailey | first1 = D. H. | author1-link = David H. Bailey (mathematician)
 | last2 = Crandall | first2 = R. E. | author2-link = Richard Crandall
 | issue = 4
 | journal = Experimental Mathematics
 | pages = 527–546
 | title = Random generators and normal numbers
 | url = http://www.emis.de/journals/EM/expmath/volumes/11/11.4/pp527_546.pdf
 | volume = 11
 | year = 2002 | doi=10.1080/10586458.2002.10504704| s2cid = 8944421 }}.
*{{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-11169-0 | zbl=1260.11001}}
*{{cite journal | zbl=0276.10028 | last=Stoneham | first=R.G. | title=On absolute $(j,ε)$-normality in the rational fractions with applications to normal numbers | journal=[[Acta Arithmetica]] | volume=22 | issue=3 | pages=277–286 | year=1973 | doi=10.4064/aa-22-3-277-286 | doi-access=free }}
*{{cite journal | zbl=0276.10029 | last=Stoneham | first=R.G. | title=On the uniform ε-distribution of residues within the periods of rational fractions with applications to normal numbers | journal=[[Acta Arithmetica]] | volume=22 | issue=4 | pages=371–389 | year=1973 | doi=10.4064/aa-22-4-371-389 | doi-access=free }}

[[Category:Eponymous numbers in mathematics]]
[[Category:Number theory]]
[[Category:Sets of real numbers]]


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