mersenneforum.org  

Go Back   mersenneforum.org > Factoring Projects > XYYXF Project

Reply
 
Thread Tools
Old 2020-08-16, 11:58   #1
JeppeSN
 
JeppeSN's Avatar
 
"Jeppe"
Jan 2016
Denmark

101000102 Posts
Question 3 as Leyland prime?

Sorry if this has been asked before.

OEIS has the following two sequences:

(A076980) Leyland numbers: 3, together with numbers expressible as n^k + k^n nontrivially, i.e., n,k > 1 (to avoid n = (n-1)^1 + 1^(n-1)).

(A094133) Leyland primes: 3, together with primes of form x^y + y^x, for x > y > 1.

Does anyone know why 3 is specifically included in these sequences? Historically, it was not. The last version of each which did not have it: A076980, version 12; A094133, version 26.

/JeppeSN

PS! There is a third sequence without the 3, which also excludes the case 2*x^x:

(A173054) Numbers of the form a^b+b^a, a > 1, b > a.
JeppeSN is offline   Reply With Quote
Old 2020-08-16, 23:26   #2
pxp
 
pxp's Avatar
 
Sep 2010
Weston, Ontario

18210 Posts
Default

Quote:
Originally Posted by JeppeSN View Post
Does anyone know why 3 is specifically included in these sequences?
The inclusion may well be my fault! With the sole intention of generating a discussion, on 6 April 2015 I wrote to the Sequence Fanatics discussion list: "The triviality condition excludes 3 (= 2^1 + 1^2), which strikes me as a useful initial term. For example, because 3 is also excluded from the Leyland primes (A094133), the comment therein that A094133 'contains A061119 as a subsequence' isn't really correct because A061119 includes 3." Unilaterally, Neil Sloane added 3 to the sequences some fifteen minutes later.

Last fiddled with by pxp on 2020-08-16 at 23:35 Reason: pluralized 'sequence'
pxp is offline   Reply With Quote
Old 2020-08-17, 09:26   #3
JeppeSN
 
JeppeSN's Avatar
 
"Jeppe"
Jan 2016
Denmark

101000102 Posts
Lightbulb

Quote:
Originally Posted by pxp View Post
The inclusion may well be my fault! With the sole intention of generating a discussion, on 6 April 2015 I wrote to the Sequence Fanatics discussion list: "The triviality condition excludes 3 (= 2^1 + 1^2), which strikes me as a useful initial term. For example, because 3 is also excluded from the Leyland primes (A094133), the comment therein that A094133 'contains A061119 as a subsequence' isn't really correct because A061119 includes 3." Unilaterally, Neil Sloane added 3 to the sequences some fifteen minutes later.
Thanks, that sounds like the explanation!

So it is n^k + k^n where either 1 < k ≤ n or 1 = k = n-1.

It is not really important if we include that exceptional case, or not.

There are similar situations for other definition, for example a generalized Fermat is b^(2^m) + 1. If you take m ≥ 0, then all numbers are generalized Fermat (which we do not want). If you take m > 0, then the classical Fermat prime F_0 is not a generalized Fermat. Finally, you can make the "mixed" criterion m > 0 or b-2 = m = 0.

/JeppeSN
JeppeSN is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Leyland Primes: ECPP proofs Batalov XYYXF Project 16 2019-08-04 00:32
On Leyland Primes davar55 Puzzles 9 2016-03-15 20:55
Leyland Numbers - Numberphile Mini-Geek Lounge 5 2014-10-29 07:28
Leyland in Popular Culture wblipp Lounge 21 2012-03-18 02:38
Paul Leyland's Mersenne factors table gone? ixfd64 Lounge 2 2004-02-18 09:42

All times are UTC. The time now is 07:23.

Wed Dec 2 07:23:30 UTC 2020 up 83 days, 4:34, 1 user, load averages: 1.07, 1.25, 1.37

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.