20150117, 18:37  #1 
May 2004
New York City
108B_{16} Posts 
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. Last fiddled with by davar55 on 20150117 at 19:23 
20150117, 21:42  #2 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
1000010110110_{2} Posts 

20150117, 22:35  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2·3·1,657 Posts 
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. Last fiddled with by Batalov on 20150117 at 23:52 Reason: tpyos 
20150117, 23:20  #4  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
26D6_{16} Posts 
Quote:
__________________ Anyway, let's redefine XYminusYX prime as a prime p = x^y  y^x, 1 < x < y. (D.Johnson, H.Lifchitz and recently N.Schneider were collecting them for years.) 2^99^2 and 2^9+9^2 are both prime. 

20150118, 04:41  #5  
May 2004
New York City
5·7·11^{2} Posts 
Quote:
Quote:
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.) 

20150118, 09:17  #6  
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
11,503 Posts 
Quote:
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 

20150211, 15:21  #7  
May 2004
New York City
5·7·11^{2} Posts 
Quote:


20150211, 15:24  #8 
May 2004
New York City
5·7·11^{2} Posts 
This was what I should have asked in the OP, without the labelled packaging.

20150211, 16:30  #9 
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
11,503 Posts 

20160315, 20:55  #10 
"Norbert"
Jul 2014
Budapest
1110110_{2} Posts 
I search PRPs of the form y^xx^y.
I made a webpage to these PRPs, you can find the page at http://primfakt.atw.hu/ Andrey Kulsha's site contains y^x+x^y PRPs, the URL is: http://www.primefan.ru/xyyxf/primes.html 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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