Well, some of us do, or pretend to, but it is not productive. Put simply it does not work. It has not ever worked, despite hundreds of documented tries (with a few tries yet to be resolved). Even the best number theorists are not sure how many there are in the interval up to p~10

^{9}, and are unsure where or when any future such discoveries will appear. See for example

https://primes.utm.edu/mersenne/index.html
See also the attachments here, the predict M45, predict M50, predict M51 or predict M52 threads, or the one based on curve fitting.

https://www.mersenneforum.org/showthread.php?t=6334 predict M45

https://www.mersenneforum.org/showth...137#post422137 predict M50

https://www.mersenneforum.org/showthread.php?t=22879 predict M51

https://www.mersenneforum.org/showthread.php?t=23892 predict M52

https://www.mersenneforum.org/showthread.php?t=24256 "Hidden number" malarky thread, based on curve fitting numerous unspecified curves

https://www.mersenneforum.org/showpo...4&postcount=69 also demonstrates the reliable failure of fits to predict even the highest known Mp on which a fit is based.

There are numerous other threads about various dubious claims to be able to make Mersenne prime predictions.

The accumulated experience of predicting or guessing Mersenne primes, in the various Predict Mxx threads, and various other threads concerning predictions, over a combined total of hundreds of guesses and predictions, is: hundreds proven composite, 6 yet to be settled, and

**zero** proven successful guesses or predictions. Another way to look at it is of over 290 guesses or predictions I've found in the forum, about 98% have been proven composite, about 2.% are yet to be determined, and

**0.00%** **(NONE) **have been proven prime. And of the as-yet-unresolved, all 6 are for exponents so large that they are not amenable to any P-1 factoring or to primality testing by PRP or LL, either within the limits of existing software capabilities or of probable hardware lifetime, so can only be attacked with trial factoring currently. Some exponents (above about 67 bits) would even have save file sizes that would exceed the capacity of currently available

**file systems**!

Six very large exponents, in exponent size order:

- https://mersenneforum.org/showpost.p...&postcount=213 p=252097800623 (the ten-billionth prime number as Mersenne exponent) is a ~37.88 bit exponent, of order 200 times larger than is practical to test on a Radeon VII GPU currently with Gpuowl in a year, and about 250 times larger than is practical currently to P-1 factor to both stages there. Some TF could be attempted with Ernst's Mfactor or Luigi's Factor5; no factor found by kriesel in Mfactor to 84 bits; kriesel continuing to 86 bits currently with 4 processes on a 4-core / 8-HT i5-1035G1;
- https://www.mersenneforum.org/showpo...33&postcount=4 p= 71324207525210468041 which is a 65.95 bit exponent, billions of times larger than what is practical to primality test or P-1 factor now; no factor in TF to 105 bits by ET_, and from 105 bits to 117 bits by kriesel using Ernst's Mfactor program; factoring from 116 bits to 117 required 27 days 13 hours with 16 processes on a dual Xeon e5-2697v2; to 118 bits is running;
- https://www.mersenneforum.org/showthread.php?t=17050, p=97600541752017987211, a 66.4 bit exponent, no factor in TF to 107.4 bits by Ernst Mayer, and from 107 bits to 118 bits by kriesel using Ernst's Mfactor program https://www.mersenneforum.org/showpo...9&postcount=18; to 119 bits is running;
- https://mersenneforum.org/showpost.p...6&postcount=10 A 75.89 bit exponent; 2
^{70237298350549551468899}-1.Trial factoring completed to 126 bits by kriesel using Ernst's Mfactor program; to 127 bits is running;
- https://www.mersenneforum.org/showthread.php?t=20238 MM127, a 127 bit exponent, p=170141183460469231731687303715884105727, no factor in TF to 185 bits by various contributors. See http://www.doublemersennes.org/mm127.php and https://mersenneforum.org/forumdisplay.php?f=99 It takes just over a month for a 5P block (5x10
^{15}) in k on a GTX1650, so going from 185 bits to 186 bits would take about 2.5 years as a continuous run in mmff. An order of magnitude faster gpu is worth about 3 bits more TF depth. MM127 is also MMM7, MMMM3, and MMMMM2. MMFF is the fastest known available software for GPU TF on modest exponent double mersenne numbers, supporting up to MM127 (or Fermat numbers to F223); newer builds at https://mersenneforum.org/showthread...=17162&page=23
- A real giant of an integer is MM82589933, with 2
^{82589933} bits. A single TF trial is about the same amount of work as primality testing M82589933, which takes most of a day on a fast gpu. TF attempts on this giant may be theoretically possible with existing software but impractically slow for mortal humans. https://www.mersenneforum.org/showpo...8&postcount=55 This is the largest, of the double Mersennes based on currently known Mersenne primes, with no known factors. As far as I could tell from the doublemersennes site, MM61 through MM82589933 (43 candidates) have no known factors yet. It is believed but not proven that none of these are prime. https://en.wikipedia.org/wiki/Double_Mersenne_number

Note these examples are well beyond the capabilities of prime95 and other primality testing or P-1 factoring software and mfaktx TF software, as well as beyond feasible primality test or P-1 run times of currently available hardware, and will remain so for a long period of time. Some will remain so forever, since their sizes dwarf the number of subatomic particles in the known universe. That makes constructing a memory of adequate size and speed for primality testing or P-1 factoring them, or sufficiently trial factoring them impossible, regardless of technical advances.

They would have extraordinarily large memory requirements. A huge extrapolation from recent gpuowl test results for stage 2 P-1 memory per buffer indicates 8.6E13 to 8.9E32 MB for 66 to 127 bit exponents. Compare to 1.6E4 MB for ram on a Radeon VII or Tesla P100 gpu.

Similarly CUDALucas file sizes are extrapolated at 8.7E12 to 2.1E31 MB for 66 to 127 bit exponents.

https://www.mersenneforum.org/showpo...1&postcount=10
Gpuowl file sizes were estimated at 8.7E12 to 2.1E31 MB for 66 to 127 bit exponents.

https://www.mersenneforum.org/showpo...7&postcount=22
Other software such as Ernst Mayer's Mfactor or Luigi Morelli's Factor5 can be used to continue TF, and in the case of MM127, George Woltman's mmff on CUDA gpus.

There are also sometimes guesses or predictions in the 300M to 900M range, that take weeks to months each to primality test on fast hardware.

The absence of correct predictions or guesses is consistent with the probability of an equal number of randomly selected prime exponents, <1ppm per guess at nontrivial exponent. Probability of primality of a number n is ~1/ln(n). Where n=2

^{p} - 1, primality probability is x~1 / ( ln(2) * p). So, for example, for p~10

^{8}, x~14ppb; p~852M, x ~ 0.000,000,001,7 (1.7 chances in a billion); for MM127, p=2

^{127}-1, x~8.5E-39.

Not really a set of predictions or claims, but more of a playful computation, is George Woltman's

computation of exponents from the Wagstaff expected mean incidence of Mersenne primes, in the second attachment. It does well at small values but quickly falls apart by p>127.

Top of reference tree:

https://www.mersenneforum.org/showpo...22&postcount=1