Leyland Primes (x^y+y^x primes)

[pxp]

Since I am no longer using the final column of my Leyland prime indexing chart to indicate intervals, I've added a "P" therein for entries that I know are proven primes. That necessitated removing the "date approximate" for Selevich's L(8656,2929) and adding a "~" in front of that entry's date, which is how I had it originally. The indices of my chart are of course one greater than the indices of the Leyland "Prime Wiki" table because they are a reflection of OEIS A094133 which has a spurious first term.

[kruoli]

Looking at that table, there are a lot of results that should be able to be proven in less than a day (at least) on "bigger" systems. Please correct me, if I am wrong here, frequent Primo users!

If correct, I'd like to reserve some of the smaller ones tomorrow for Primo. I will specify them futher then and also double check them on FactorDB.

In the list of kar_bon, maybe we should try to certify the "orange" entries, too?

[xilman]

Originally Posted by bur

Impressive compilation. Do you have data on which of the numbers are just PRPs? It would make a nice list of candidates for primo.

Not sure whether you realise this, but that's the reason why these numbers have been singled out and given a particular name.

Long ago I pointed out that they seem to have a reasonable density of primes of all sizes, have a very simple algebraic description and have no obvious properties which can be exploited by special-purpose algorithms.

Several have been used to set records for the size of a certified prime. They are frequently used, I believe, to test new implementations of general primality testing algorithms.

[kruoli]

Originally Posted by kruoli

If correct, I'd like to reserve some of the smaller ones tomorrow for Primo. I will specify them futher then and also double check them on FactorDB.

Excuse me for the delay. For now, I'd like to reserve only http://www.rieselprime.de/ziki/Leyland_prime_P_3222_553 and watch how long it takes on my machine. I guess that's better before I get overboard. After that finished, I'll report.

[Yusuf]

I found a new record for largest Leyland PRP
It is 386642 digits long (previous record is 386434 digits)
PRP: 81650^54369+54369^81650

[kruoli]

Originally Posted by kruoli

After that finished, I'll report.

It took around 18-20 h on a 5950X (all cores used). The certificate is uploaded. Who has to be informed to update the table? I guess we can wait with this until the numbers below are finished as well.

I'd like to reserve

• \(L(3107, 712)\)
• \(L(3710, 303)\)
• \(L(5041, 70)\)
• \(L(3099, 1078)\)
• \(L(3782, 315)\)
• \(L(3784, 315)\)

for Primo certification next. This are the six smallest Leyland PRPs (unproven) currently.

[pxp]

Originally Posted by kruoli

Who has to be informed to update the table?

I will update my table as your results come in.

[kruoli]

My reservations from above are now completed and are currently being processed by FactorDB. Thus, I'd like to reserve:

• \(L(3178, 1005)\)
• \(L(3185, 1182)\)
• \(L(3171, 1256)\)
• \(L(3081, 1568)\)
• \(L(3070, 1781)\)
• \(L(4504, 165)\)
• \(L(6886, 29)\)

Since I already submitted some proofs to FactorDB for numbers which have been proven before independently but had no certificate on FactorDB, as soon as the above are completed, all Leyland primes below 10,000 digits will be certified in FactorDB. I checked this with a script; if I made some error, please correct me! Edit: There are three certificates missing which I cannot upload currently. Sorry.

As a personal side note, the last number in the list above will be my first 10k digit ECPP run. Knowing that this is not something impressive at all by itself, I am still pleased.

PS: I just saw FactorDB got a hardware upgrade. They now run:

• CPU: AMD Ryzen 3700X
• RAM: 2x16 GB
• SSD: 3x1 TB RAID 5

[NorbSchneider]

Another new PRP:
457^60454+60454^457, 160803 digits.

[pxp]

Originally Posted by pxp

I started (yesterday) the interval from L(299999,10) to L(300999,10). A preliminary estimate suggests that this will require some two-and-a-half months.

I'm done. In addition to Anatoly Selevich's L(314738,9), I found 16 new primes. The two-and-a-half months was pretty close. So to get up to 305000 decimal digits (which is on my schedule) will take another ten months. But first I want to make sure that there are no primes in the interval between Sergey Batalov's L(328574,15) and Yusuf AttarBashi's L(81650,54369).

[pxp]

Originally Posted by pxp

I want to make sure that there are no primes in the interval between Sergey Batalov's L(328574,15) and Yusuf AttarBashi's L(81650,54369).

In fact, I found two new primes in that interval:

<386642> Jun 2021 L(81650,54369)
<386561> Aug 2021 L(80565,62824) new
<386548> Aug 2021 L(83747,41272) new
<386434> May 2014 L(328574,15)

This means that these four (currently largest) Leyland PRPs are consecutive. As an informal proof thereof I have saved outputs of the pfgw testing as summarized here.