The world's largest known prime number has 41,024,320 digits.

That's so big it doesn't make sense to even attempt to write it down. Fortunately, it's a Mersenne prime, which means it can be written more simply with an equation with the following form:

M(p) = 2^p - 1

Where p, the exponent in the formula, is also a prime number. For the new world's largest prime number, p = 136,279,841. Here's Matt Parker announcing the discovery of the new Mersenne prime on 21 October 2024 in a 10-minute video and presenting all of its 41,024,320 digits:

To get a sense of how big this number is, we created the following chart showing the exponents of each of the known Mersenne primes as of 23 January 2025 on the vertical axis plotted against their rank from lowest to highest on the horizontal axis. The newest Mersenne prime, M(136,279,841) is shown in red.

Two things immediately stand out on the cart. First, it's a true hockey stick chart, thanks to the exponential increase in the Mersenne prime exponents. Second, there is an exceptionally large gap between the previous record-holder for largest known prime number M(82,589,933) at Rank 51 and the newest Mersenne prime M(136,279,841) that is currently at Rank 52.

We can de-hockey stick the chart in one of two ways. We can either convert the vertical axis to be in logarithmic scale or we can express the exponents as their natural logs. In the second chart, we've taken the second route in part because it makes the numbers easier to handle.

When we present the Mersenne prime exponents this way, they appear to follow a straight line pattern, which is interesting. In our third chart, we did some basic linear regression analysis to tease out the overall relationship between the size of the Mersenne prime exponents and their rank, which is really telling us about how far apart the exponents are from each other.

The dashed line represents the overall trend for the increasing size of the known Mersenne prime exponents, which seems to fit the data fairly well. But when we look back at the second chart, we can't help but see what looks like a change in the space between exponents that kicks in after they get past Rank 40. Specifically, once we're looking at Mersenne prime exponents above Rank 40, there is less space between them than what our initial linear regression suggests there should be.

We wondered if there might be a second pattern in these figures, so we generated two new linear regressions. One based on the ten Mersenne prime exponents from Rank 42 through 51 (omitting the newest one since our first chart suggests it may be an outlier) and another based on the Mersenne prime exponents from Rank 1 through 42. The fourth chart presents our results:

This fourth chart strongly suggests there may be something to our observations. Mersenne prime exponents above M(25,964,951) at Rank 42 appear to have branched off from the overall trend we see for Mersenne prime exponents from Rank 1 through 42 and may possibly have branched off earlier with M(20,996,011) at Rank 40. But as we're about to show, that may not the only branch because that pattern appears to repeat. In the fifth chart, we've added several parallel branches that pass in close proximity to many of the lower ranked Mersenne prime exponents.

If this pattern reliably repeats, it would provide a means by which we could reasonably predict where to find the "next" Mersenne prime exponents on a given branch. For example, if the newest, largest Mersenne prime M(136,279,841) is on such a branch, the next larger Mersenne prime exponent might be found lurking among potential prime exponents somewhere between it and a possible upper end of 155,645,033, with a exponent of 142,143,581 falling in the middle of a likely range in which it might be found.

Going back to our observation that there appears to be an exceptionally large gap between the Mersenne primes corresponding to M(82,589,933) at Rank 51 and the newest Mersenne prime M(136,279,841) currently at Rank 52, the branching pattern we've identified suggests there may be as many as two additional Mersenne prime in between them. The first could potentially fall between the potential prime exponents of 87,747,713 and 103,582,291 and the second could fall between the potential prime exponents of 100,504,603 and 118,641,197. The final chart shows where these 'next' Mersenne prime numbers might be if they fall on the same branch defined by the spacing between Mersenne prime exponents from Rank 42 through Rank 51. Note that the current Rank 52 Mersenne prime M(136,279,841) would move out to Rank 54 in this scenario.

Then again, the large gap in the value of the Mersenne Prime exponents may be a sign that the Mersenne prime exponents have reached the end of a branch and will have more distance between them. At least, until a new branch might become evident.

It's quite possible the scenarios we described for locating the next Mersenne primes are reasonable. At this writing, there are still thousands of potential Mersenne prime exponent candidates within the target ranges we've identified whose status has not yet been determined. The only thing we reasonably know for certain is that there will be another Mersenne prime identified someday that's larger than M(136,279,841).

In terms of potential, should the existence of these branching patterns pan out and be useful, they would make it possible for future Mersenne primes to be discovered in clusters, with several found within a relatively short period after each new Mersenne prime is confirmed. With luck, it won't take another six years to locate the next Mersenne prime.