Leyland in Popular Culture
 

From James Gleick's new book [I]The Information[/I] page 339

[B][INDENT]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[sup]y[/sup] + y[sup]x[/sup]).[/INDENT][/B]

I knew about Paul's involvement in these, but I wasn't aware they were called Leyland numbers. There is [URL="http://en.wikipedia.org/wiki/Leyland_number"]Wikipedia article[/URL] with a first draft of 2006.

Well Spotted William
 

Or was that the Lesser?

Was Popular Culture some "in" Hasselhon reference?

David

[QUOTE=wblipp;293156]From James Gleick's new book [I]The Information[/I] page 339

[B][INDENT]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[sup]y[/sup] + y[sup]x[/sup]).[/INDENT][/B]

I knew about Paul's involvement in these, but I wasn't aware they were called Leyland numbers. There is [URL="http://en.wikipedia.org/wiki/Leyland_number"]Wikipedia article[/URL] with a first draft of 2006.[/QUOTE]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

Silly Me
 

[QUOTE=xilman;293180]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[/QUOTE]

There I was thinking it was about an old car I used to have.

What is known, or otherwise conjectured, about Leyland numbers L = [TEX]x^y+y^x[/TEX] = [TEX]v^w+w^v[/TEX] which can be expressed in the form in more than one way? Are there none, finitely many or infinitely many? Are any examples known?

[QUOTE=Brian-E;293228]What is known, or otherwise conjectured, about Leyland numbers L = [TEX]x^y+y^x[/TEX] = [TEX]v^w+w^v[/TEX] which can be expressed in the form in more than one way? Are there none, finitely many or infinitely many? Are any examples known?[/QUOTE]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

[QUOTE=xilman;293230]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[/QUOTE]

Small correction: (2,3) = 17 = (1,[B]16[/B]). 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.

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"][/code] 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.

[QUOTE=Brian-E;293228]What is known, or otherwise conjectured, about Leyland numbers L = [TEX]x^y+y^x[/TEX] = [TEX]v^w+w^v[/TEX] which can be expressed in the form in more than one way? Are there none, finitely many or infinitely many? Are any examples known?[/QUOTE]

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.

[QUOTE=wblipp;293243]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.[/QUOTE]

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"))))[/CODE]

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).

[QUOTE=wblipp;293243]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.[/QUOTE]
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.:rolleyes:

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