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

[pxp]

Originally Posted by bur

3^125330 + 125330^3
The certificate is uploaded and verified to/by factordb.

Thanks for this. I try to keep the proven-primes column on my Leyland-primes listing up-to-date but unless it falls just after the contiguous initial stretch I would need a heads-up of its status to notice. At the moment, all Leyland primes < 13300 digits are factor-db proven. Of the larger numbers, 2929^8656+8656^2929 (noted with a K instead of a P) still remains without a certificate at factordb.

[frmky]

20018^63+63^20018 is also prime.

[pxp]

Originally Posted by lghu

My 'Leyland-1M' project found this PRP:
211185^54364+54364^211185 is Fermat and Lucas PRP!
1000027 digits, index: 21589915517 (if my program is correct).

It seems that this is the only Leyland PRP with at least one million decimal digits, but fewer than one million one hundred decimal digits.

[lghu]

Originally Posted by pxp

It seems that this is the only Leyland PRP with at least one million decimal digits, but fewer than one million one hundred decimal digits.

This is not so surprising, since for example there is no Leyland PRP between L(238176,19) and L(65073,48202) [digits 304569 and 304742].

[pxp]

Originally Posted by lghu

This is not so surprising, since for example there is no Leyland PRP between L(238176,19) and L(65073,48202) [digits 304569 and 304742].

Since we know that there are 63 Leyland PRPs from digits 300000 to 305000, we can say that, in that range, on average there is one PRP every 80 digit-lengths. I had guessed that in the greater-than-one-million-digits range there might be one PRP every 200 digit-lengths, so the surprise really was that there is a solution at all. If I may ask, how many candidates did you look at before finding your 1000027-digit PRP?

[lghu]

Originally Posted by pxp

If I may ask, how many candidates did you look at before finding your 1000027-digit PRP?

I can't say exactly. To test with the same digits, approx. 600-700 Fermat-tests are needed after searching for small prime divisors.

[NorbSchneider]

Another new PRP:
21650^43269+43269^21650, 187591 digits.

[lghu]

A new PRP:

101645^94522+94522^101645, 505739 digit, index: 6214332272

[lghu]

A new 1M+ digits PRP:

218767^37314+37314^218767, 1000175 digits, index: 21595765797

[pxp]

Originally Posted by lghu

218767^37314+37314^218767, 1000175 digits, index: 21595765797

Is your search technique exhaustive? By which I mean, are you able to tell us — now or at some point in the future — that this 1000175-digit result and your previous 1000027-digit result are consecutive PRPs? Or are you just picking random Leyland number pairs constrained by digit size?

[bur]

Originally Posted by paulunderwood

There have been a few recent ECPP numbers not making their respective top20 tables. I told Prof Caldwell. I guess he is busy retiring. Perhaps someone can email him about these numbers.

The Top 20 has been updated now. Impressive how the sizes increased with Andreas' open source implementation.