[seqfan] Re: Leyland Numbers

Neil Sloane njasloane at gmail.com
Mon Apr 6 18:31:02 CEST 2015


I made some changes to A96980.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Mon, Apr 6, 2015 at 12:16 PM, Hans Havermann <gladhobo at teksavvy.com>
wrote:

> https://oeis.org/A076980
>
> "numbers expressible as n^k + k^n nontrivially, i.e. n,k > 1 (to avoid n =
> (n-1)^1 +1^(n-1))"
>
> 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.
>
> I'm also looking at Alonso del Arte's 2006 comment that "Crandall &
> Pomerance named these numbers in honor of Paul Leyland, in reference to
> 2638^4405 + 4405^2638, the largest known prime of this form." I've had a
> look at Crandall & Pomerance's 2005 "Prime Numbers: A Computational
> Perspective" and the reference is:
>
> "A sensational announcement in July 2004 by Franke, Kleinjung, Morain, and
> Wirth is that, thanks to fastECPP, the Leyland number 4405^2638 +
> 2638^4405, having 15071 decimal digits, is now proven prime."
>
> It strikes me that they are called Leyland numbers *in spite of* the
> primality proof so I find this reference a little misleading. At any rate,
> there's a couple of larger proven-prime examples now known.
>
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