Gillies' conjecture

{{expert-subject|1=Mathematics|date=January 2014|reason=Needs to be checked by editor with advanced mathematics knowledge}}

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{{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

| doi-access = free

}} in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good{{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

| doi-access = free

}} and Daniel Shanks.{{cite book

|last= Shanks

|first= Daniel

|year= 1962

|title= Solved and Unsolved Problems in Number Theory

|publisher= Spartan Books

|location= Washington

|pages=198

}} The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted (but also open) Lenstra–Pomerance–Wagstaff conjecture.

The conjecture

: $\text{If }$ $A < B < \sqrt{M_p}\text{, as }B/A\text{ and }M_p \rightarrow \infty\text{, the number of prime divisors of }M$

: $\text{ in the interval }[A, B]\text{ is Poisson-distributed with}$

:: $\text{mean }\sim
\begin{cases}
\log(\log B /\log A) & \text{ if }A \ge 2p\\
\log(\log B/\log 2p) & \text{ if } A < 2p
\end{cases}$

He noted that his conjecture would imply that

The number of Mersenne primes less than $x$ is $~\frac{2}{\log 2} \log\log x$ .
The expected number of Mersenne primes $M_p$ with $x \le p \le 2x$ is $\sim2$ .
The probability that $M_p$ is prime is $~\frac{2 \log 2p }{p\log 2}$ .

Incompatibility with Lenstra–Pomerance–Wagstaff conjecture

The Lenstra–Pomerance–Wagstaff conjecture gives different values:{{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

| doi-access = free

}}Chris Caldwell, [http://primes.utm.edu/mersenne/heuristic.html Heuristics: Deriving the Wagstaff Mersenne Conjecture]. Retrieved on 2017-07-26.

The number of Mersenne primes less than $x$ is $~\frac{e^\gamma}{\log 2} \log\log x$ .
The expected number of Mersenne primes $M_p$ with $x \le p \le 2x$ is $\sim e^\gamma$ .
The probability that $M_p$ is prime is $~\frac{e^\gamma\log ap}{p\log 2}$ with a = 2 if p = 3 mod 4 and 6 otherwise.

Asymptotically these values are about 11% smaller.

Results

While Gillie's conjecture remains open, several papers have added empirical support to its validity, including Ehrman's 1964 paper.{{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

| doi-access = free

}}

References

Category:Conjectures

Category:Unsolved problems in number theory

Category:Hypotheses

Category:Mersenne primes