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Old 2012-03-16, 02:08   #1
wblipp
 
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Default 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.
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Old 2012-03-16, 07:20   #2
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Default 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.
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Old 2012-03-16, 10:36   #3
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Quote:
Originally Posted by wblipp View Post
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
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Old 2012-03-16, 11:27   #4
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Originally Posted by xilman View Post
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.
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Old 2012-03-16, 19:49   #5
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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?
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Old 2012-03-16, 20:02   #6
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Originally Posted by Brian-E View Post
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
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Old 2012-03-16, 20:59   #7
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Quote:
Originally Posted by xilman View Post
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.
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Old 2012-03-16, 21:36   #8
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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
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Old 2012-03-16, 22:11   #9
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Quote:
Originally Posted by Brian-E View Post
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.
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Old 2012-03-16, 22:29   #10
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Quote:
Originally Posted by wblipp View Post
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).
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Old 2012-03-16, 22:36   #11
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Quote:
Originally Posted by wblipp View Post
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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