en>Anders9ustafsson: Added link to LINCOA C# source code

2014-02-03T21:06:22Z

Added link to LINCOA C# source code

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In [[number theory]], '''Gillies' conjecture''' is a [[conjecture]] about the distribution of prime divisors of [[Mersenne numbers]] and was made by [[Donald B. Gillies]] in a 1964 paper<ref>{{cite journal
| author = Donald B. Gillies
| title = Three new Mersenne primes and a statistical theory
| journal = Mathematics of Computation
| volume = 18
| pages = 93–97
| year = 1964
| doi = 10.1090/S0025-5718-1964-0159774-6
| issue = 85
}}</ref> in which he also announced the discovery of three new [[Mersenne prime]]s. The conjecture is a specialization of the [[prime number theorem]] and is a refinement of conjectures due to [[I. J. Good]]<ref>{{cite journal
| author = I. J. Good
| title = Conjectures concerning the Mersenne numbers
| journal = Mathematics of Computation
| volume = 9
| pages = 120–121
| year = 1955
| doi = 10.1090/S0025-5718-1955-0071444-6
| issue = 51
}}</ref> and [[Daniel Shanks]].<ref>{{cite book
|last= Shanks
|first= Daniel
|year= 1962
|title= Solved and Unsolved Problems in Number Theory
|publisher= Spartan Books
|location= Washington
|pages=198
}}</ref> The conjecture remains an open problem, although several papers have added empirical support to its validity.

==The conjecture==
:<math>\text{If }</math><math>A < B < \sqrt{M_p}\text{, as }B/A\text{ and }M_p \rightarrow \infty\text{, the number of prime divisors of }M</math>
:<math>\text{ in the interval }[A, B]\text{ is Poisson-distributed with}</math>
::<math>
\text{mean }\sim
\begin{cases}
\log(\log B /\log A) & \text{ if }A \ge 2p\\
\log(\log B/\log 2p) & \text{ if } A < 2p
\end{cases}
</math>

He noted that his conjecture would imply that
# The number of Mersenne primes less than <math>x</math> is <math>~\frac{2}{\log 2} \log\log x</math>.
# The expected number of Mersenne primes <math>M_p</math> with <math>x \le p \le 2x</math> is <math>\sim2</math>.
# The probability that <math>M_p</math> is prime is <math>~\frac{2 \log 2p }{p\log 2}</math>.

==Known results==
While Gillie's conjecture remains an open problem, several papers have added empirical support to its validity, including Ehrman's 1964 paper<ref>{{cite journal
| author = John R. Ehrman
| title = The number of prime divisors of certain Mersenne numbers
| journal = Mathematics of Computation
| volume = 21
| pages = 700–704
| year = 1967
| doi = 10.1090/S0025-5718-1967-0223320-1
| issue = 100
}}</ref> as well as Wagstaff's 1983 paper.<ref>{{cite journal
| author = Samuel S. Wagstaff
| title = Divisors of Mersenne numbers
| journal = Mathematics of Computation
| volume = 40
| pages = 385–397
| year = 1983
| doi = 10.1090/S0025-5718-1983-0679454-X
| issue = 161
}}</ref>

==Notes==
{{reflist}}

{{DEFAULTSORT:Gillies Conjecture}}
[[Category:Number theory]]
[[Category:Conjectures]]
[[Category:Hypotheses]]

{{Numtheory-stub}}

LINCOA - Revision history

en>Anders9ustafsson: Added link to LINCOA C# source code