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

[kar_bon]

I've made some changes/approvements in the Wiki for Leyland primes/PRPs:

The latest PRP can be found here
- added a history entry to give the post# with date
- the "date found" from FactorDB
- not yet Leyland# filled

There's also a "Proved" history entry possible: see here
- "Prove date" and "Program" from FactorDB
- History date from forum post with link

There're also different categories for proven primes and PRP. Because of sorting by digits, the smallest unproven number can be found as first entry here = Leyland prime P 3147 214 = 7334 digits.
This can help to identify and prove smaller PRPs for others.
In the table view the numbers which marked "proven" but no proof is available in FactorDB are marked.

Further there's a page which creates a CSV format to copy/paste for further use.

I'm now going to insert more/all known Leyland numbers in the wiki with data I've found:
- proven dates/names from the other thread (most from RichD) and found dates from here
- dates from old primes from Leyland page as "When reserved"/"When completed".

[kar_bon]

I don't know if this was shown somewhere but found nothing, so here's the current distribution of known Leyland primes/PRP's for x<40000 and y<25000:

[pxp]

I recognize those 100000-digit numbers on the right. :) I played in Mathematica with a similar graph last night with the idea of superimposing curves of 60000-, 80000-, and 100000-digit x^y+y^x but I couldn't even figure out how to generate those curves. :/

[pxp]

Originally Posted by pxp

I played in Mathematica with a similar graph last night with the idea of superimposing curves of 60000-, 80000-, and 100000-digit x^y+y^x but I couldn't even figure out how to generate those curves.

I finally kludged something together (ContourPlot was the Mathematica function that I was missing). I'm not sure if the waves on the end of the <60000> curve are real or an artifact.

[xilman]

Originally Posted by pxp

I finally kludged something together (ContourPlot was the Mathematica function that I was missing). I'm not sure if the waves on the end of the <60000> curve are real or an artifact.

An interesting plot. I think the waves are very likely an artefact.

Above about, say, 8000,the distribution of the points looks to me very much like a uniformly random sample of the triangle. Presumably it is not, or the distribution would look random at the lower regions as well.

It may be interesting to apply a scaling to the Y values, such that the populated area becomes square and then to investigate the hypothesis that the (x,y) co-ordinates are drawn independently from a uniform random distribution. If the likelihood is significantly different try to discover a distribution which better matches the observations.

Any takers? I'm not sure my statistics ability is (yet) up to the task.

(added in edit: Henry,this seems like an area where you have some expertise.)

[pxp]

Originally Posted by xilman

I think the waves are very likely an artefact...

Indeed. I've fixed that, fixed the aspect ratio to make the x=y line bisect the axes, made the points smaller, and added two greenish curves to indicate the interval that I am currently exploring (I should be done in ten days).

[pxp]

Originally Posted by pxp

That makes L(32907,92) #1296.

I have examined all Leyland numbers in the nine gaps between L(32907,92) <64623>, #1296, and L(29934,157) <65733> and found 22 new primes. That makes L(29934,157) #1327.

[pxp]

I have written a blog article regarding my ambitious Leyland-prime search schedule for the next two years.

[NorbSchneider]

Another new PRP:
7257^17528+17528^7257, 67672 digits.

[pxp]

Originally Posted by pxp

That makes L(29934,157) #1327.

I have examined all Leyland numbers in the gap between L(29934,157) <65733>, #1327, and L(40182,47) <67189> and found 20 new primes. That makes L(40182,47) #1348 and advances the index to L(31870,131), #1354.

[NorbSchneider]

Another new PRP:
12511^17556+17556^12511, 71933 digits.