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 2012-03-16, 02:08 #1 wblipp     "William" May 2003 New Haven 2×32×131 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 xy + yx). 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.
 2012-03-16, 07:20 #2 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts Well Spotted William Or was that the Lesser? Was Popular Culture some "in" Hasselhon reference? David Last fiddled with by davieddy on 2012-03-16 at 07:25 Reason: Might have misplaced the "David" there.
2012-03-16, 10:36   #3
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

280216 Posts

Quote:
 Originally Posted by wblipp 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 xy + yx). 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.
For some reason C&P named them after me in their second edition. It was months, or possibly years, before I found out about it.

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

2012-03-16, 11:27   #4
davieddy

"Lucan"
Dec 2006
England

2·3·13·83 Posts
Silly Me

Quote:
 Originally Posted by xilman For some reason C&P named them after me in their second edition. It was months, or possibly years, before I found out about it. 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
There I was thinking it was about an old car I used to have.

 2012-03-16, 19:49 #5 Brian-E     "Brian" Jul 2007 The Netherlands 2×23×71 Posts What is known, or otherwise conjectured, about Leyland numbers L = $x^y+y^x$ = $v^w+w^v$ which can be expressed in the form in more than one way? Are there none, finitely many or infinitely many? Are any examples known?
2012-03-16, 20:02   #6
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

240028 Posts

Quote:
 Originally Posted by Brian-E What is known, or otherwise conjectured, about Leyland numbers L = $x^y+y^x$ = $v^w+w^v$ which can be expressed in the form in more than one way? Are there none, finitely many or infinitely many? Are any examples known?
Good question, and not one I've seen asked before. There is at least one case, (2,3) = (1,4), but I doubt there are many more. I'd be very surprised if there are infinitely many and, off the top of my head, can't think of any other examples.

Paul

2012-03-16, 20:59   #7
axn

Jun 2003

13×192 Posts

Quote:
 Originally Posted by xilman Good question, and not one I've seen asked before. There is at least one case, (2,3) = (1,4), but I doubt there are many more. I'd be very surprised if there are infinitely many and, off the top of my head, can't think of any other examples.l
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 non-trivial solution exists.

 2012-03-16, 21:36 #8 science_man_88     "Forget I exist" Jul 2009 Dumbassville 26·131 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"] where the first number in the quotes for each is x^y+y^x mod 6 the others are x mod 6 and y mod 6 respectfully this is all that seems to happen for x and y under 250 does this help for them to be prime. Last fiddled with by science_man_88 on 2012-03-16 at 21:38
2012-03-16, 22:11   #9
wblipp

"William"
May 2003
New Haven

235810 Posts

Quote:
 Originally Posted by Brian-E What is known, or otherwise conjectured, about Leyland numbers L = $x^y+y^x$ = $v^w+w^v$ which can be expressed in the form in more than one way? Are there none, finitely many or infinitely many? Are any examples known?
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.

2012-03-16, 22:29   #10
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts

Quote:
 Originally Posted by wblipp 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.
leyland numbers + primes

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"))))
is what I have working for me so far to try finding prime leyland numbers ( yes I know efforts are likely under-way already, and that the biggest one confirmed so far is x=2638;y=4405).

2012-03-16, 22:36   #11
Brian-E

"Brian"
Jul 2007
The Netherlands

326610 Posts

Quote:
 Originally Posted by wblipp 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.
Yes, if anyone does discover any of these numbers - despite Paul's and axn's pessimism - then Leyland Taxi Numbers would be a very good name for them. We'd soon educate Google about what it means.

@Science Man You've heard the famous story about Hardy visiting Ramanujan in hospital, haven't you?

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