Puzzle
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Ponder This Challenge - April 2010 - Hex = dec, removing last 21 digits

Ponder This Challenge:

This month's challenge is based on a suggestion from Leo Broukhis.

The prime number 4535653, when translated to base 16, gives the hexadecimal number 0x453565, which has the same digits as the original number, omitting the last digit.

Find another example of a prime number that, translated to hexadecimal, yields the same digits, omitting the last 21 digits.

If you like this challenge, you may find April's riddle on UYHIP (link resides outside of ibm.com) interesting as well.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

  • Our original solution was as follows:

    If the digits are x_i, then we have that \sum_{i=0}^N 10^i x_i = \sum_{i=21}^N 16^{i-21} x_i. Defining d_i = 10^{i+21} - 16^i, we get a lattice problem that can easily be solved by backtracking. 15 out of the 16 solutions are divided by 2, 3, or 7, leaving only 228076225315225436100944141061067721086496831646429721832862692146190993521
    57014522058098996385254903281545695887189603267201 as a possible candidate. Primality checking verifies that it indeed is a valid solution.

    We received the following elegant solution from one of our solvers:

    Preliminary estimates show that such a number would have 125 decimal places. Using random seeds, look for fixpoints of the Transformation T: "interpret n_{10}, last 21 digits dropped, as (T(n))_{16}", until you hit a prime.

Solvers

  • Puzzlet Chung (03/31/2010 05:07 AM EDT)
  • (03/31/2010 05:07 AM EDT)
  • Lukasz Bolikowski (03/31/2010 06:36 AM EDT)
  • (03/31/2010 06:36 AM EDT)
  • Dan Dima (03/31/2010 03:05 PM EDT)
  • (03/31/2010 03:05 PM EDT)
  • Joseph DeVincentis (03/31/2010 04:20 PM EDT)
  • (03/31/2010 04:20 PM EDT)
  • (04/01/2010 05:57 PM EDT)
  • Michael Liepelt (04/01/2010 05:57 PM EDT)
  • (04/01/2010 05:57 PM EDT)
  • Chuck Carroll (04/01/2010 11:00 PM EDT)
  • (04/01/2010 11:00 PM EDT)
  • Xuhui Li (04/02/2010 02:05 AM EDT)
  • (04/02/2010 02:05 AM EDT)
  • Victor A Chang (04/02/2010 04:40 AM EDT)
  • (04/02/2010 04:40 AM EDT)
  • Christian Blatter (04/02/2010 05:23 AM EDT)
  • (04/02/2010 05:23 AM EDT)
  • Thomas Jessen (04/02/2010 07:15 AM EDT)
  • (04/02/2010 07:15 AM EDT)
  • Clive Tong (04/02/2010 11:07 AM EDT)
  • (04/02/2010 11:07 AM EDT)
  • Andreas Eisele (04/02/2010 11:25 AM EDT)
  • (04/02/2010 11:25 AM EDT)
  • Michael Quist (04/02/2010 01:30 PM EDT)
  • (04/02/2010 01:30 PM EDT)
  • Arthur Breitman (04/02/2010 02:23 PM EDT)
  • (04/02/2010 02:23 PM EDT)
  • Andreas Stiller (04/02/2010 07:06 PM EDT)
  • Emile Joubert (04/02/2010 09:22 PM EDT)
  • (04/02/2010 09:22 PM EDT)
  • Harald Bögeholz (04/04/2010 11:59 AM EDT)
  • (04/04/2010 11:59 AM EDT)
  • Mark Mixer (04/04/2010 04:16 PM EDT)
  • (04/04/2010 04:16 PM EDT)
  • Daniel Bitin (04/04/2010 04:58 PM EDT)
  • (04/04/2010 04:58 PM EDT)
  • John T. Robinson (04/04/2010 10:46 PM EDT)
  • (04/04/2010 10:46 PM EDT)
  • Kang Seonghoon (04/05/2010 02:41 AM EDT)
  • (04/05/2010 02:41 AM EDT)
  • Kevin Williamson (04/05/2010 07:54 AM EDT)
  • (04/05/2010 07:54 AM EDT)
  • Adam Daire (04/05/2010 07:05 PM EDT)
  • (04/05/2010 07:05 PM EDT)
  • (04/07/2010 10:21 AM EDT)
  • Albert Stadler (04/07/2010 10:21 AM EDT)
  • (04/07/2010 10:21 AM EDT)
  • Joseph Bonneau (04/07/2010 02:51 PM EDT)
  • (04/07/2010 02:51 PM EDT)
  • Piotr Komisarski (04/07/2010 03:58 PM EDT)
  • (04/07/2010 03:58 PM EDT)
  • Eugene Vasilchenko (04/08/2010 11:48 AM EDT)
  • (04/08/2010 11:48 AM EDT)
  • Phil Muhm (04/08/2010 03:24 PM EDT)
  • Reiner Martin (04/08/2010 07:40 PM EDT)
  • Jan van Delden (04/09/2010 07:53 AM EDT)
  • Donald T. Dodson (04/09/2010 01:52 PM EDT)
  • Chuck Grant (04/11/2010 11:40 PM EDT)
  • Pataki Gergely (04/13/2010 06:45 PM EDT)
  • (04/13/2010 06:45 PM EDT)
  • Lachezar Hristov (04/14/2010 05:54 AM EDT)
  • (04/14/2010 05:54 AM EDT)
  • Sylvain Becker (04/14/2010 06:08 PM EDT)
  • Nagy Zoltan (04/17/2010 08:03 AM EDT)
  • Sevag Papazian (04/17/2010 08:42 AM EDT)
  • Wolfgang Kais (04/19/2010 12:34 PM EDT)
  • (04/19/2010 12:34 PM EDT)
  • Venkata Ramayya Muramalla (04/19/2010 01:43 PM EDT)
  • (04/19/2010 01:43 PM EDT)
  • Bojan Basic (04/19/2010 08:13 PM EDT)
  • (04/19/2010 08:13 PM EDT)
  • Sidharth Lath (04/25/2010 04:01 PM EDT)
  • (04/25/2010 04:01 PM EDT)
  • Frank Morten (04/25/2010 04:52 PM EDT)
  • (04/25/2010 04:52 PM EDT)
  • Stephen Herbert (04/27/2010 11:57 PM EDT)
  • (04/27/2010 11:57 PM EDT)
  • Rani Hod (04/28/2010 06:07 AM EDT)
  • (04/28/2010 06:07 AM EDT)
  • Yuval Filmus (04/28/2010 09:38 PM EDT)
  • (04/28/2010 09:38 PM EDT)
  • Mauro Bampo (04/29/2010 06:23 PM EDT)
  • Don Ryan (04/29/2010 10:52 PM EDT)

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