On Leyland Primes

davar55 · 2015-01-17, 18:37

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 Leyland-alpha 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 Leyland-beta prime.

Find primes p that can be expressed as both an L-alpha and L-beta prime simultaneously (with different x,y).
Find primes p that can be expressed in at least two different ways as L-alpha or L-beta primes (with different x,y).

I don't currently know if there are any.

TimSorbet · 2015-01-17, 21:42

Quote:

Originally Posted by davar55

Let's call a prime p that can be written in the form p = x^y - y^x > 0 for integral x,y a Leyland-beta prime.

I'm not sure this is a valid definition.

Batalov · 2015-01-17, 22:35

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 "Leyland-beta primes" or "Capitalism" or "Cosmology"). Never mind that they come out poorly thought out (and hence self-contradictory), 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 dead-end).
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.

Batalov · 2015-01-17, 23:20

Quote:

Originally Posted by davar55

Let's call a prime p that can be written in the form p = x^y - y^x > 0 for integral x,y a Leyland-beta prime.

Every prime is a "Leyland-beta prime", because p = (p+1)^1 - 1^(p+1)! Hooray!
__________________

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.)

2^9-9^2 and 2^9+9^2 are both prime.

davar55 · 2015-01-18, 04:41

Quote:

Originally Posted by Batalov

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 "Leyland-beta primes" or "Capitalism" or "Cosmology"). Never mind that they come out poorly thought out (and hence self-contradictory), 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 dead-end).
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:

Originally Posted by Batalov

Every prime is a "Leyland-beta prime", because p = (p+1)^1 - 1^(p+1)! Hooray!
__________________

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.)

2^9-9^2 and 2^9+9^2 are both prime.

(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 Leyland-like primes, not
an x,y that produced two different primes (which is a valid additional question, so thanks for an example.)

xilman · 2015-01-18, 09:17

Quote:

Originally Posted by Batalov

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.)

Thanks for that link.

Since the New Year I've been factoring x^y-y^x < 2^1024 where 1<x<y. There are 17289 of them, including the two special cases of 2^3-3^2 and 2^4-4^2. 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

davar55 · 2015-02-11, 15:21

Quote:

Originally Posted by xilman

Thanks for that link.
Since the New Year I've been factoring x^y-y^x < 2^1024 where 1<x<y. There are 17289 of them, including the two special cases of 2^3-3^2 and 2^4-4^2. 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.

We're interested in these primes, and factorizations. Progress report?

davar55 · 2015-02-11, 15:24

Quote:

Originally Posted by davar55

In any case, I was looking for two different representations of a prime p as Leyland-like primes, not
an x,y that produced two different primes (which is a valid additional question, so thanks for an example.)

This was what I should have asked in the OP, without the labelled packaging.

xilman · 2015-02-11, 16:30

Quote:

Originally Posted by davar55

We're interested in these primes, and factorizations. Progress report?

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.

NorbSchneider · 2016-03-15, 20:55

I search PRPs of the form y^x-x^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^x-x^y PRPs exists much more than y^x+x^y PRPs.
For example to x= 5000 894 y^x-x^y and 426 y^x+x^y PRPs,
x=10000 1530 y^x-x^y and 787 y^x+x^y PRPs.
I have all the y^x-x^y PRPs to x=10800, and a few for higher x values.
Are you interesting to join me searching the y^x-x^y PRPs?

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2015-01-17, 18:37	#1
davar55 May 2004 New York City 5·7·11² 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 Leyland-alpha 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 Leyland-beta prime. Find primes p that can be expressed as both an L-alpha and L-beta prime simultaneously (with different x,y). Find primes p that can be expressed in at least two different ways as L-alpha or L-beta primes (with different x,y). I don't currently know if there are any. Last fiddled with by davar55 on 2015-01-17 at 19:23

2015-01-17, 22:35	#3
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 89×113 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 "Leyland-beta primes" or "Capitalism" or "Cosmology"). Never mind that they come out poorly thought out (and hence self-contradictory), 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 dead-end). 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 2015-01-17 at 23:52 Reason: tpyos

2016-03-15, 20:55	#10
NorbSchneider "Norbert" Jul 2014 Budapest 7C₁₆ Posts	I search PRPs of the form y^x-x^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^x-x^y PRPs exists much more than y^x+x^y PRPs. For example to x= 5000 894 y^x-x^y and 426 y^x+x^y PRPs, x=10000 1530 y^x-x^y and 787 y^x+x^y PRPs. I have all the y^x-x^y PRPs to x=10800, and a few for higher x values. Are you interesting to join me searching the y^x-x^y PRPs?