# FAQ: Where is the Next Mersenne Prime?

### By Chris Caldwell

[ See also: Deriving the Wagstaff Mersenne Conjecture ]

Where is the next Mersenne prime? To answer this question, let
M_{n} be the *n*th Mersenne
prime. Take a careful look at the graph of log_{2}(log_{2}(M_{n}))
versus *n*. (That is, the logarithm base two of the
logarithm base two of the *n*th Mersenne.)

One can not miss that this graph is amazingly linear. Perhaps
just as amazing is that this is just what one would predict using simple
heuristic arguments. In 1964 Gillies made a conjecture implying
the slope is 1/2 [Gillies64]. Unfortunately
his heuristic contradicts the prime number theorem. In 1980 both
Lenstra and Pomerance independently conjectured there are asymptotically
(e* ^{gamma}*/log 2) log
log

*x*Mersenne primes less than

*x*[Pomerance81]. In what follows we will address this conjecture in the tripartite form given by Wagstaff a couple years later [Wagstaff83]:

- The number of Mersenne primes less than or equal to
*x*is about (e/log 2) * log log^{gamma}*x*. (Again*gamma*is Euler's constant). - The expected number of Mersenne primes 2
-1 with^{p}*p*between*x*and 2*x*is about e.^{gamma} - The
probability that 2
-1 is prime is about (e^{p}log^{gamma}*ap*)/(*p*log 2) where*a*=2 if*p*=3 (mod 4) and*a*=6 if*p*=1 (mod 4).

This means that the geometric mean of the ratio of two successive Mersenne
exponents is 2 raised to 1/e* ^{gamma }*or about 1.47576. The
fact this is close to 3/2 is probably the reason Eberhart (and many
others after him) looked at his limited data and guessed that the ratio
is exactly 3/2. This does appeal to those with neo-Pythagoristic
tendencies (those who think numbers should be rational), but has very
little other argument in favor of it.

## Empirical Data from the First Mersennes

Let's compare this conjecture to our graph: if it is correct, then the
distribution of the log of the log of the Mersennes is a Poisson process
and if we graph log_{2}(log_{2}(M_{n}))
versus *n* we
should get approximately a straight line with slope 1/e* ^{gamma }*(about
0.56145948). Indeed, using the first 51 (known) Mersennes we get
a regression line with slope 0.5366 and a correlation coefficient

*R*

^{2}= 0.9885. Not too bad.

The gaps in any Poisson process should also exhibit specific behaviors.
For example, the list of the *n*th gaps should be uncorrelated (we
get *R* = -0.222 for the first 50 gaps). Also their mean and standard
deviation should be near the parameter for the distribution (we get
0.531 and 0.459 respectively, relatively near the parameter 1/e* ^{gamma }*=
0.561...).

Further, the cumulative distribution of the gaps between values of a
Poisson process should be governed by an exponential distribution.
In particular, the probability density f(*t*) for the gaps and the
probability p(*t*) the gap length is as follows [Ross93,
p215].

(Usually the parameter *t* is a Poisson process is time, here it
is log_{2}(log_{2}(M_{n})).)

Noll's nebulous "Island Theory" for Mersennes (which I have never seen quantified) seems to roughly state that Mersennes occur in clumps with gaps between them. Perhaps so, but that is exactly what you would expect from any Poisson process!

Next let's check the predicted Poisson gap distribution against the gaps between the known Mersennes. In the graph below I plot the observed cumulative frequencies of these gaps (red points) versus the expected distribution (brown curve) for the known Mersenne gaps. Again the match is excellent for the relatively few data points we have! (See [Schroeder83] for a similar analysis of the first 28 Mersennes.)

Finally, if this distribution is Poisson with mean e* ^{gamma}* and we view log

_{2}(log

_{2}(M

_{n}))as our 'time' scale we expect about 1.78 Mersennes per interval. Dividing the log

_{2}(log

_{2}(M

_{n})) scale so far explored into 27 intervals: [0,1), [1,2), ..., [26,27) and counting the number of Mersennes in each we get the following.

Mersennes | Observed | Expected |
---|---|---|

0 | 3 | 4.5 |

1 | 8 | 8.1 |

2 | 8 | 7.2 |

3 | 6 | 4.3 |

4+ | 2 | 2.9 |

A goodness-of-fit test then gives us a Chi-squared value of 1.54 and
a *p-value* of about 0.82. This could provide weak evidence that this is a Poisson with
mean e* ^{gamma}* well. (More technically: if it were such a distribution,
then the probability of seeing a Chi-squared value this much larger
than 0 due to "sampling error" alone would be about 82%)

## What about the next Mersenne?

So what can we gather from this? One thing is that given one Mersenne
prime exponent's *p*, the next one will fall, on the average, near
1.47576*p.* But not too near the average much of the time--sometimes
the gaps will be small, sometimes large. So the next larger Mersenne
exponent might be over 85,000,000 yielding a Mersenne with nearly
26 million digits. Or it may not.

Lets be more specific graphically. **If** we had no idea what the
first Mersennes were, we would predict values for *t*=log_{2}(log_{2}(M_{n}))
as follows.

But we do know 51 Mersennes. Suppose they are the first 51 (that
is, there are no primes below the 51*st* that lies undiscovered--remember
not all of these numbers have been tested yet). So if those we
know are the first 51, then we need only look at the expected gap
(Poisson processes essentially "start over" after each event).
Now we get the much narrower graphs:

## Where do these conjectures come from?

They come from Wagstaff's article: [Wagstaff83]. But that answer is not much help is it? So we created a separate page deriving the Wagstaff Mersenne conjectures.

For fun let's end with two more graphs: the probability density graph
for the *n*th Mersenne, and the probability that it occurs before a given *t *= log_{2}(log_{2}(M_{n}))--both
assuming no knowledge of previous Mersennes.