Here, large is defined as an exponent beyond the mersenne.org limit, currently 10

^{9}, which is almost 2

^{30}
Before considering larger exponents, it's useful to consider the scale of effort needed near the limit of

mersenne.org.

TF to recommended bit levels for near the 10

^{9} limit takes days per exponent on the fastest gpus and best software.

P-1 testing to recommended bounds near the 10

^{9} limit takes ~4 days per exponent in GpuOwL on fast gpus such as Radeon VIIs.

Primality testing near the 10

^{9} limit takes around 6 months per exponent with GpuOwL on a Radeon VII. Prime95 benchmarks indicate about 13 months each primality test near 10

^{9} on a Xeon Phi 7250 (68 cores 1 worker at ~1.4-1.5 GHz clock and 16GB 7200MHz MCDRAM in same chip package).

**ONLY** PRP should be considered for such long tests. (LL testing lacks the Gerbicz error check in all existing software and lacks even the weak Jacobi symbol check in some software, so is unlikely to complete correctly for very long runs.)

Primality testing effort scales as roughly exponent

^{2.1}. P-1 effort and TF effort scale similarly, with a smaller proportionality constant, ~1/40 that of primality testing, each. Conversely, the probability of finding a prime diminishes per test as exponent increases.

Software support for large exponents is limited, and the more so as exponent increases. There is little reason to exert effort in software development for large exponents soon, since the remaining work in mersenne.org will take more than a century at currently projected rates.

https://www.mersenneforum.org/showpo...5&postcount=11
A summary of software support is available in the attachment at

http://www.mersenneforum.org/showpos...91&postcount=2
Interim residues for a limited wide ranging variety of fft lengths and specified exponents are available at

https://www.mersenneforum.org/showpo...4&postcount=12
https://www.mersenne.ca/ covers status for exponents up to 10

^{10}, and coordinates trial factoring for exponents over 10

^{9} up to 2

^{32}.

Gigadigit Mersennes are covered in subforum Operation Billion Digits

https://mersenneforum.org/forumdisplay.php?f=50.

Assignment reservation and result submission page, and display of status for trial factoring is at

https://www.mersenne.ca/obd
I do not know of any software currently capable of P-1 factoring such large numbers in a reasonable amount of time or with acceptable reliability. See

https://mersenneforum.org/showthread.php?t=24486 for discussion of run time scaling and requirements. It's likely Mlucas will have sufficient

P-1 capability implemented as a byproduct of the coming F33 attempt.

Gigadigit primality tests are technically possible now in Mlucas or certain versions of Gpuowl, but the duration per test on available hardware is longer than the likely hardware lifetime.So the hardware would need to be replaced and work in progress migrated, and great care exerted with backups of work in progress and error checking along the way.

There's a forum thread regarding 10-gigadigit numbers, in which even larger are also mentioned. Only trial factoring is feasible for these. There's little or no point in doing so, other than as an amusement.

https://mersenneforum.org/showthread.php?t=16801
Double Mersennes: For Mersenne numbers with Mersenne numbers as exponents, see the Operazione Doppi Mersennes subforum:

https://mersenneforum.org/forumdisplay.php?f=99. Also

http://www.doublemersennes.org/ There's a thread discussing when it might be feasible to primality test MM61. Assuming Moore's law continues, over 100 years. But Moore's law is slowing and faces hard physics limits.

https://mersenneforum.org/showthread.php?t=21522
Some of the challenges and changes required are discussed briefly in

https://mersenneforum.org/showthread.php?t=17354 and

https://mersenneforum.org/showpost.p...6&postcount=16
There's also a bit of discussion of the difficulty of even attempting TF on the triple Mersenne MMM127.

https://mersenneforum.org/showthread.php?t=18522
Top of reference tree:

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