On Leyland Primes
Let's call a prime p that can be written in the form p = x^y + y^x for integral x > y > 1 (or x=y=1 or x=2,y=1) a Leylandalpha prime.
Let's call a prime p that can be written in the form p = x^y  y^x > 0 for integral x,y a Leylandbeta prime. Find primes p that can be expressed as both an Lalpha and Lbeta prime simultaneously (with different x,y). Find primes p that can be expressed in at least two different ways as Lalpha or Lbeta primes (with different x,y). I don't currently know if there are any. 
[QUOTE=davar55;392707]Let's call a prime p that can be written in the form [B]p = x^y  y^x > 0[/B] for integral x,y a Leylandbeta prime.[/QUOTE]
I'm not sure this is a valid definition. :huh: 
There, in a nutshell, lies the usual problem with davar55's constructions.
David's Rule #1: Never mind existing definitions. David's Rule #2: Make up your own definitions (be that "Leylandbeta primes" or "Capitalism" or "Cosmology"). Never mind that they come out poorly thought out (and hence selfcontradictory), and in the end define an empty set (or everything). David's Rule #3: Start making up theories. Never look back at definitions or attempt to fix them and start over (even if getting into a deadend). David's Rule #4: When others stop reading at the level of shoddy definitions, argue that they are too lazy to "read the whole thing". ......... PROFIT!!! I.e. in this case, an endless supply of troll food. Our best guess is that getting that food is the point of the exercise. 
[QUOTE=davar55;392707]Let's call a prime p that can be written in the form p = x^y  y^x > 0 for integral x,y a Leylandbeta prime.
[/QUOTE] [B]Every prime[/B] is a "Leylandbeta prime", because p = (p+1)^1  1^(p+1)! Hooray! __________________ Anyway, let's redefine [URL="http://www.primenumbers.net/prptop/searchform.php?form=x^yy^x&action=Search"]XYminusYX prime[/URL] as a prime p = x^y  y^x, [B]1 < x < y[/B]. (D.Johnson, H.Lifchitz and recently N.Schneider were collecting them for years.) 2^99^2 and 2^9+9^2 are both prime. 
[QUOTE=Batalov;392729]There, in a nutshell, lies the usual problem with davar55's constructions.
David's Rule #1: Never mind existing definitions. David's Rule #2: Make up your own definitions (be that "Leylandbeta primes" or "Capitalism" or "Cosmology"). Never mind that they come out poorly thought out (and hence selfcontradictory), and in the end define an empty set (or everything). David's Rule #3: Start making up theories. Never look back at definitions or attempt to fix them and start over (even if getting into a deadend). David's Rule #4: When others stop reading at the level of shoddy definitions, argue that they are too lazy to "read the whole thing". ......... PROFIT!!! I.e. in this case, an endless supply of troll food. Our best guess is that getting that food is the point of the exercise.[/QUOTE] [QUOTE=Batalov;392739][B]Every prime[/B] is a "Leylandbeta prime", because p = (p+1)^1  1^(p+1)! Hooray! __________________ Anyway, let's redefine [URL="http://www.primenumbers.net/prptop/searchform.php?form=x^yy^x&action=Search"]XYminusYX prime[/URL] as a prime p = x^y  y^x, [B]1 < x < y[/B]. (D.Johnson, H.Lifchitz and recently N.Schneider were collecting them for years.) 2^99^2 and 2^9+9^2 are both prime.[/QUOTE] (a) There's nothing wrong with my "definitions" in cosmology, though I was willing to discuss them with anyone who offered better ones. I never called anyone here "lazy" for not reading anything of mine. I did suggest that some here criticized without even reading the monograph. I stand by that. (b) The explanations I gave re capitalism were never formal definitions, just foundational ideas. The connection between Capitalism and Freedom is certainly not merely mine. (c) I did err here in this thread in the beta case,forgetting the special case you cited. Acknowledged. This happens to me perhaps more than others here (though there have been plenty of errors by others in presenting puzzles) because (c1) I tend to offer more puzzles than most, and (c2) I'm too often in a hurry to post when I get a new idea. My bad. In any case, I was looking for two different representations of a prime p as Leylandlike primes, not an x,y that produced two different primes (which is a valid additional question, so thanks for an example.) 
[QUOTE=Batalov;392739]Anyway, let's redefine [URL="http://www.primenumbers.net/prptop/searchform.php?form=x^yy^x&action=Search"]XYminusYX prime[/URL] as a prime p = x^y  y^x, [B]1 < x < y[/B]. (D.Johnson, H.Lifchitz and recently N.Schneider were collecting them for years.)[/QUOTE]Thanks for that link.
Since the New Year I've been factoring x^yy^x < 2^1024 where 1<x<y. There are 17289 of them, including the two special cases of 2^33^2 and 2^44^2. Right now 8948 of them remain unfactored. Factors are still showing up at the rate of about 150 per day. Once the rate settles down a little I'll make the tables available and invite others to contribute  some time in February I guess. Paul 
[QUOTE=xilman;392764]Thanks for that link.
Since the New Year I've been factoring x^yy^x < 2^1024 where 1<x<y. There are 17289 of them, including the two special cases of 2^33^2 and 2^44^2. Right now 8948 of them remain unfactored. Factors are still showing up at the rate of about 150 per day. Once the rate settles down a little I'll make the tables available and invite others to contribute  some time in February I guess. [/QUOTE] We're interested in these primes, and factorizations. Progress report? 
[QUOTE=davar55;392751]
In any case, I was looking for two different representations of a prime p as Leylandlike primes, not an x,y that produced two different primes (which is a valid additional question, so thanks for an example.)[/QUOTE] This was what I should have asked in the OP, without the labelled packaging. 
[QUOTE=davar55;395190]We're interested in these primes, and factorizations. Progress report?[/QUOTE]Factors still coming in at >50 per day. More than 6300 composites remaining. All composites >C90. All SNFS candidates >= 145 digits. Approximately one t30 test by ECM.

I search PRPs of the form y^xx^y.
I made a webpage to these PRPs, you can find the page at [url]http://primfakt.atw.hu/[/url] Andrey Kulsha's site contains y^x+x^y PRPs, the URL is: [url]http://www.primefan.ru/xyyxf/primes.html[/url] y^xx^y PRPs exists much more than y^x+x^y PRPs. For example to x= 5000 894 y^xx^y and 426 y^x+x^y PRPs, x=10000 1530 y^xx^y and 787 y^x+x^y PRPs. I have all the y^xx^y PRPs to x=10800, and a few for higher x values. Are you interesting to join me searching the y^xx^y PRPs? 
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