Ponder This Challenge - January 2017 - Rational solutions for x**y=y**x

Ponder This Challenge:

There is exactly one integer solution for 0<x<y, xy=yx.

Hertsel Amrani (thanks!) asked me about rational solutions and inspired the idea for this month's challenge, which is to find a rational solution for xy=yx, where the last eight digits of x are different from each other.

Supply your answer as a formula for x and y, and the first and last 10 digits of x and y.

Solution

• Click here to view the solution

  The general solution for x**y=y**x is x = ((n+1)/n)**n and y = ((n+1)/n)**(n+1).

  For a finite decimal representation, n should be 2**k*5**l.

  n=80 is the simplest solution; n=3125 is a solution for the bonus '*'. Note that we need to switch x>y for that.

Solvers

• *Robert Gerbicz (01/01/2017 01:21 PM IDT)
• Jonathan Hoch (01/01/2017 03:31 PM IDT)
• Kang Jin Cho (01/01/2017 04:48 PM IDT)
• *Kipp Johnson (01/01/2017 07:10 PM IDT)
• *Arthur Vause (01/01/2017 09:38 PM IDT)
• *Yan-Wu He (01/01/2017 09:41 PM IDT)
• *George W Barnett (01/01/2017 11:42 PM IDT)
• *Jesse Rearick (02/01/2017 01:32 AM IDT)
• *Tamir Ganor & Shouky Dan (02/01/2017 09:37 AM IDT)
• *Raymond Lo (02/01/2017 10:27 AM IDT)
• Wilder Boyden (02/01/2017 10:30 AM IDT)
• Shirish Chinchalkar (02/01/2017 02:52 PM IDT)
• *Serge Batalov (03/01/2017 08:05 AM IDT)
• *Zusheng Ji (03/01/2017 08:27 AM IDT)
• James Dow Allen (03/01/2017 03:27 PM IDT)
• Alex Fleischer (03/01/2017 04:37 PM IDT)
• *Yuanjie Pu (04/01/2017 04:26 AM IDT)
• *Liubing Yu (04/01/2017 06:03 AM IDT)
• John Snyder (04/01/2017 07:11 PM IDT)
• *Reiner Martin (05/01/2017 01:51 AM IDT)
• *Radu-Alexandru Todor (05/01/2017 01:53 AM IDT)
• *Luke Pebody (05/01/2017 06:56 AM IDT)
• *Mathias Schenker (05/01/2017 06:52 PM IDT)
• Rudy Cortembert (05/01/2017 07:47 PM IDT)
• *Philippe Fondanaiche (06/01/2017 04:12 PM IDT)
• *Stephen Lesko (06/01/2017 07:14 PM IDT)
• *Aviv Nisgav (08/01/2017 05:39 AM IDT)
• *Franciraldo Cavalcante (08/01/2017 08:27 PM IDT)
• *Yves Miennée (08/01/2017 10:01 PM IDT)
• *Peter Gerritson (08/01/2017 10:27 PM IDT)
• *Tyler Mullen (09/01/2017 08:47 AM IDT)
• *Adrian Neacsu (09/01/2017 11:36 AM IDT)
• Martí Bosch and Sebastian Bohm (09/01/2017 01:28 PM IDT)
• *JJ Rabeyrin (10/01/2017 12:06 AM IDT)
• *Francis Golding (10/01/2017 04:43 PM IDT)
• *Oleg Vlasii (11/01/2017 03:32 PM IDT)
• *Muralidhar Seshadri (13/01/2017 09:18 AM IDT)
• Joaquim Neves Carrapa (13/01/2017 05:04 PM IDT)
• Husen Wang (13/01/2017 11:49 PM IDT)
• *Motty Porat (15/01/2017 01:20 AM IDT)
• David F.H. Dunkley (15/01/2017 12:20 PM IDT)
• *Aaradesh (16/01/2017 05:58 PM IDT)
• Hyung Sik Hwang (17/01/2017 07:34 AM IDT)
• *Robert Lang (17/01/2017 07:02 PM IDT)
• *David Greer (17/01/2017 08:39 PM IDT)
• *Pål Hermunn Johansen (18/01/2017 12:50 AM IDT)
• *Andreas Stiller (18/01/2017 07:21 PM IDT)
• *Steven Langerwerf (19/01/2017 03:54 PM IDT)
• Dan Dima (20/01/2017 11:44 AM IDT)
• Jeff Boyle (23/01/2017 02:34 AM IDT)
• Adir HaCohen (23/01/2017 02:56 AM IDT)
• The Questionable Squad (23/01/2017 07:03 AM IDT)
• *Armin Krauss (25/01/2017 08:39 PM IDT)
• *Joehyun Kim (25/01/2017 08:52 PM IDT)
• *Sergey Koposov (26/01/2017 09:47 PM IDT)
• *Chris Shannon (27/01/2017 03:49 AM IDT)
• *Chuck Carroll (27/01/2017 02:22 PM IDT)
• *Vladimir M. Milovanović (27/01/2017 03:11 PM IDT)
• Hao Zheng (27/01/2017 04:34 PM IDT)
• *Sergey Grishaev (27/01/2017 09:34 PM IDT)
• Pieter Meijer (27/01/2017 11:14 PM IDT)
• *Alessandro Marzo (28/01/2017 07:28 PM IDT)
• *Bert Dobbelaere (29/01/2017 08:19 PM IDT)
• Jason Lee, Jamie Jorgensen & Peter Farbman (30/01/2017 08:53 PM IDT)
• *Teake Nutma (30/01/2017 09:39 PM IDT)
• *Uoti Urpala (31/01/2017 05:25 PM IDT)
• *Martin Sergio H. Fæster (01/02/2017 11:54 PM IDT)