Re: [math-fun] Leyland primes
Hans Havermann recently mentioned Leyland primes (primes in the form x^y + y^x), so I looked them up on OEIS. I expected to see a bunch of random-looking odd numbers. And for the most part, that's what I saw. But I noticed that some of them have remarkably long runs of 0s. For instance the 30th term is a 465-digit decimal number, of which 305 consecutive digits are 0. And the 37th term is a 613-digit number of which 417 consecutive digits are 0. Why should that be? And does something similar happen in other radices? Thanks. One possibility is that in those cases y^x is much greater than x^y and y is divisible by 10. If so, I'd expect that I'd find primes in the form x^y - y^x to often have long runs of 9s. I found a list at A123206, and see that that is indeed the case. So I guess that's the explanation.
Keith: "One possibility is that in those cases y^x is much greater than x^y and y is divisible by 10." Indeed, your examples of the 30th and 37th Leyland primes are (x,y) = (357,20) and (471,20) while the 18th and 34th Leyland primes, (81,80) and (237,200), do not exhibit the long string of zeros. I recently conjectured that for d > 11, 10^(d-1) + (d-1)^10 is the smallest (base ten) d-digit Leyland number. After a bit these will exhibit a long string of zeros after the initial 1. Alas, no primes with y = 10 are yet known.
Does the OEIS have a mode that lets you display the numbers in a different base? That might bring out some patterns that are not visible from the default decimal presentation of entries. I looked at the homepage and the Welcome page but didn’t see anything relevant. (Of course, it is trivial to copy and paste sequences into an application that can handle base conversion.) While we’re talking nonstandard modes of presentation of numbers: Have any mathematical discoveries been triggered by the graphical or auditory presentations of OEIS sequences? Jim Propp On Monday, June 4, 2018, Hans Havermann <gladhobo@bell.net> wrote:
Keith: "One possibility is that in those cases y^x is much greater than x^y and y is divisible by 10."
Indeed, your examples of the 30th and 37th Leyland primes are (x,y) = (357,20) and (471,20) while the 18th and 34th Leyland primes, (81,80) and (237,200), do not exhibit the long string of zeros.
I recently conjectured that for d > 11, 10^(d-1) + (d-1)^10 is the smallest (base ten) d-digit Leyland number. After a bit these will exhibit a long string of zeros after the initial 1. Alas, no primes with y = 10 are yet known.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Hi Jim!
Does the OEIS have a mode that lets you display the numbers in a different base? Answer: No, sorry, that's not a feature we offer
While we’re talking nonstandard modes of presentation of numbers: Have any mathematical discoveries been triggered by the graphical or auditory presentations of OEIS sequences?
Answer: Yes, for the graphical display, many! Example: see the paper we wrote about the Yellowstone Permutation (A098550). 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@gmail.com On Mon, Jun 4, 2018 at 10:06 AM, James Propp <jamespropp@gmail.com> wrote:
Does the OEIS have a mode that lets you display the numbers in a different base? That might bring out some patterns that are not visible from the default decimal presentation of entries.
I looked at the homepage and the Welcome page but didn’t see anything relevant. (Of course, it is trivial to copy and paste sequences into an application that can handle base conversion.)
While we’re talking nonstandard modes of presentation of numbers: Have any mathematical discoveries been triggered by the graphical or auditory presentations of OEIS sequences?
Jim Propp
On Monday, June 4, 2018, Hans Havermann <gladhobo@bell.net> wrote:
Keith: "One possibility is that in those cases y^x is much greater than x^y and y is divisible by 10."
Indeed, your examples of the 30th and 37th Leyland primes are (x,y) = (357,20) and (471,20) while the 18th and 34th Leyland primes, (81,80) and (237,200), do not exhibit the long string of zeros.
I recently conjectured that for d > 11, 10^(d-1) + (d-1)^10 is the smallest (base ten) d-digit Leyland number. After a bit these will exhibit a long string of zeros after the initial 1. Alas, no primes with y = 10 are yet known.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Hans Havermann -
James Propp -
Keith F. Lynch -
Neil Sloane