20120316, 02:08  #1 
"William"
May 2003
New Haven
2340_{10} Posts 
Leyland in Popular Culture
From James Gleick's new book The Information page 339
The number 593 is more interesting than it looks; it happens to be the sum of nine squared and two to the ninth  thus a "Leyland number" (any number than can be expressed as x^{y} + y^{x}). I knew about Paul's involvement in these, but I wasn't aware they were called Leyland numbers. There is Wikipedia article with a first draft of 2006. 
20120316, 07:20  #2 
"Lucan"
Dec 2006
England
6,451 Posts 
Well Spotted William
Or was that the Lesser?
Was Popular Culture some "in" Hasselhon reference? David Last fiddled with by davieddy on 20120316 at 07:25 Reason: Might have misplaced the "David" there. 
20120316, 10:36  #3  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
23575_{8} Posts 
Quote:
I think it was because I had plugged them as fine candidates for general primality proving software. They are reasonably common at all sizes, they have a simple algebraic form and they do not appear to have any simple algebraic structure which presently known special purpose algorithms can exploit. Paul 

20120316, 11:27  #4  
"Lucan"
Dec 2006
England
6,451 Posts 
Silly Me
Quote:


20120316, 19:49  #5 
"Brian"
Jul 2007
The Netherlands
110011000000_{2} Posts 
What is known, or otherwise conjectured, about Leyland numbers L = = which can be expressed in the form in more than one way? Are there none, finitely many or infinitely many? Are any examples known?

20120316, 20:02  #6  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
11·919 Posts 
Quote:
Paul 

20120316, 20:59  #7 
Jun 2003
3^{2}·523 Posts 
Small correction: (2,3) = 17 = (1,16). A quick search for 1 <= x < y <= 100 shows only trivial solutions (where one of them is 1). I doubt if a nontrivial solution exists.

20120316, 21:36  #8 
"Forget I exist"
Jul 2009
Dumbassville
8,369 Posts 
I've been doing some math in PARI:
Code:
["1,0,1", "1,0,5", "1,1,0", "1,3,4", "1,4,3", "1,5,0", "5,1,4", "5,2,3", "5,3,2", "5,4,1", "5,4,5", "5,5,4"] Last fiddled with by science_man_88 on 20120316 at 21:38 
20120316, 22:11  #9 
"William"
May 2003
New Haven
2^{2}×3^{2}×5×13 Posts 
When I googled for Leyland Numbers, the Google search bar helpfully suggested I might want Leyland Taxi Numbers. I thought these  or perhaps only the smallest of these  would be Leyland Taxi Numbers. Google thought it would be a phone number for a Leyland Taxi Service.

20120316, 22:29  #10  
"Forget I exist"
Jul 2009
Dumbassville
20261_{8} Posts 
Quote:
or leyland numbers + math should stop the confusion. for those interested: Code:
for(x=2,100,a=x%6;forstep(y=if(a%3==1,x+3,x+1),100,if(a==0  a==3  a==5,[4,2],if(a==1,[2,4],if(a==2,6,if(a==4,[2,2,2])))),if(isprime(x^y+y^x),print(x","y" is prime")))) 

20120316, 22:36  #11  
"Brian"
Jul 2007
The Netherlands
CC0_{16} Posts 
Quote:
@Science Man You've heard the famous story about Hardy visiting Ramanujan in hospital, haven't you? 

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