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

[pxp]

Originally Posted by XYYXF

There are small gaps up to (20956,283). That milestone seems to be reachable :-)

Absolutely. And there's no reason to stop there! Which accounts for my current engagement with two gaps beyond that. ;-)

[pinhodecarlos]

Originally Posted by rogue

FYI, I'm making a small change to the PRPNet server code so that server stats use y^x-x^y for the - form and x^y+y^x for the + form.

Who will host the PRPNet server so I can point my cores to it?

[rogue]

Originally Posted by pinhodecarlos

Who will host the PRPNet server so I can point my cores to it?

I am not hosting a public server as I run one on my network at home. It is easy enough to set up your own server if you are interested.

[pxp]

L(14670,3083)

[NorbSchneider]

I reached x=12,940 and found 1 new PRP:
12174^12937+12937^12174, 52854 digits.

Hans, how calculate you the Leyland# to a given (x,y) pair?
For example to (18661,390) how calculate you the Leyland# 90659013?
Which program/code does it? I would use this to the y^x-x^y PRPs also.

[pxp]

L(13495,5154)

[pxp]

Originally Posted by NorbSchneider

Hans, how calculate you the Leyland# to a given (x,y) pair?

When I started this about a year ago, I created a database of the first 331682621 Leyland numbers by their (x,y) designation. The hard part was getting them sorted by size (there are lots of near-equal bunches). In actual use I have broken that up into five (slightly overlapping) parts. So when I go through a gap I'm actually testing each (x,y) pair in sequence by its Leyland# index and when I find a prime I have the Leyland# directly.

The database couldn't tell me the Leyland# indices of the largest nine currently-known pairs so I wrote a program (in Mathematica — all my primality testing is done in Mathematica as well) that would do that. As with the previously-mentioned sorting problem there's a difficulty in balancing the needs of speed and accuracy but I eventually got the program where I felt it was working well and correctly, testing it on numbers in my database.

Since then, every time a new Leyland prime is discovered I run it through that Mathematica program (even though for my finds I already know the index).

[NorbSchneider]

I reached x=12,970 and found 1 new PRP:
10821^12968+12968^10821, 52317 digits.

Mark the y^x-x^y PRPs page is updated, the new PRPs from you and me
are on the page now.

Hans, the Leyland# to a given (x,y) pair determine you also with a database
and a Mathematica program, thank for sharing this. I try to write a program
in C# to determine the "Leyland#" to the y^x-x^y PRPs.

[pxp]

Originally Posted by NorbSchneider

I try to write a program in C# to determine the "Leyland#" to the y^x-x^y PRPs.

You might want to match your indices to this OEIS list.

[pxp]

L(13137,5242)

[NorbSchneider]

I reached x=12,985 and found 1 new PRP:
11434^12977+12977^11434, 52664 digits.