Ponder This Challenge - March 2019 - Rational approximation of square root

Every real number can be approximated as a ratio of two integers. For example, pi is close to 22/7 but even closer to 3138/999; and there is an even better approximation with a denominator < 1000 (what is it?).

We define m=BAT(x,N) as the denominator, m, of a fraction k/m which is the best approximation for x with denominator < N.

Let N1 be the number of electrons in one coulomb of electric charge.

Let N2 be energy (measured in electron volts) released in the fusion of 1,000 deuterium atoms and 1,000 tritium atoms to form 1,000 He-4 atoms and 1,000 neutrons.

Find two non-square integers n1 and n2 such that b1=BAT(sqrt(n1),N1)=BAT(sqrt(n2),N1) and b2=BAT(sqrt(n1),N2)=BAT(sqrt(n2),N2) and both b1 and b2 are prime numbers.

Bonus question: for the smallest solution that also satisfy BAT(sqrt(n1),10000)=BAT(sqrt(n2),10000), what is the connection between BAT(sqrt(n1),10000) and IBM? Hint: see April 2011's challenge.

Solution

• Click here to view the solution

  N1 is approximately 6.24E18 and N2 is 1.76E10. The solution we aimed for is n1=53 and n2=477, where BAT(sqrt(n1),N1)=BAT(sqrt(n1),N1)=5929223246159194801 and BAT(sqrt(n1),N2)=BAT(sqrt(n1),N2)=17217982601, both of which are prime numbers.
  The answer to the bonus question is BAT(sqrt(n1),10000)=BAT(sqrt(n2),10000)=9100, which is the number of patents IBM had in 2018.

  Uoti Urpala sent us a really nice solution, which does not require more than a pen and paper:
  n1 = 10000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000001
  n2 = 10000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000002
  (first integers after (1e50 + 0.5)**2)
  BAT(sqrt(n), L) will be 2 for all L up to about 1e49, as the square roots are very close to 1e50 + 1/2.

  Oscal Volpatti sent us a solution for all the last 26 years of IBM patents numbers:
  year 1993: 40610066,162440264
  year 1994: 82555214,742996926
  year 1995: 63363131,253452524
  year 1996: 6894703,27578812
  year 1997: 976405495,8787649455
  year 1998: 2842190725,71054768125
  year 1999: 47036688,423330192
  year 2000: 251952448,6298811200
  year 2001: 44692204,178768816
  year 2002: 608115615,15202890375
  year 2003: 16982390,67929560
  year 2004: 95759246,861833214
  year 2005: 17758607,71034428
  year 2006: 15182737,60730948
  year 2007: 16727176,66908704
  year 2008: 108575937,977183433
  year 2009: 303117418,7577935450
  year 2010: 230388515,2073496635
  year 2011: 177366413,8690954237
  year 2012: 442656442,3983907978
  year 2013: 10627194,42508776
  year 2014: 83928763,755358867
  year 2015: 3248882,12995528
  year 2016: 568907081,14222677025
  year 2017: 17107172,68428688
  year 2018: 53,477

Solvers

• Anders Kaseorg (28/02/19 03:15 AM IDT)
• *Harald Bögeholz (03/03/19 01:40 AM IDT)
• Todd Will (03/03/19 05:33 PM IDT)
• Uoti Urpala (04/03/19 04:48 AM IDT)
• *Clive Tong (04/03/19 09:42 PM IDT)
• *Dominik Reichl (04/03/19 09:42 PM IDT)
• *Eden Saig (05/03/19 12:15 AM IDT)
• *Deron Stewart (06/03/19 01:42 AM IDT)
• *Daniel Chong Jyh Tar (07/03/19 12:43 AM IDT)
• *Michael Rosola (06/03/19 08:53 PM IDT)
• *Andreas Stiller (07/03/19 12:36 PM IDT)
• Dan Dima (07/03/19 04:39 PM IDT)
• *Kang Jin Cho (09/03/19 06:50 PM IDT)
• *Deniz Yuret (09/03/19 09:50 PM IDT)
• *Haye Chan (10/03/19 02:46 PM IDT)
• *JJ Rabeyrin (10/03/19 09:12 PM IDT)
• *Alper Halbutogullari (11/03/19 01:58 AM IDT)
• *Arthur Vause (11/03/19 07:52 PM IDT)
• *Shouky Dan & Tamir Ganor (11/03/19 12:53 PM IDT)
• Tom Chen (12/03/19 04:36 AM IDT)
• Graham Hemsley (13/03/19 10:41 AM IDT)
• Hugo Pfoertner (15/03/19 05:42 PM IDT)
• *Reda Kebbaj (16/03/19 10:01 PM IDT)
• *Daniel Bitin (17/03/19 06:26 PM IDT)
• *David Greer (17/03/19 11:39 PM IDT)
• *Adam Daire (19/03/19 07:40 PM IDT)
• Thomas Egense (19/03/19 09:48 AM IDT)
• Chris Shannon (21/03/19 12:03 AM IDT)
• Sanandan Swaminathan (21/03/19 05:26 AM IDT)
• *Jeewoo Lee (23/03/2019 04:36 PM IDT)
• *Kelvin Kim (23/03/2019 05:33 PM IDT)
• *Joonsoo Lee (23/03/2019 06:36 PM IDT)
• *Min Jung Kim (24/03/2019 04:59 PM IDT)
• Bryce Fore (24/03/2019 08:58 PM IDT)
• Vladimir Volevich (25/03/2019 01:12 PM IDT)
• *Chuck Carroll (26/03/19 05:53 AM IDT)
• *Marco Bellocchi (26/03/2019 11:54 PM IDT)
• *Armin Krauß (28/03/2019 01:08 PM IDT)
• *Jiho Lee (30/03/2019 07:51 AM IDT)
• Kenneth Kwok (30/03/2019 07:09 PM IDT)
• *Dieter Beckerle (30/03/2019 11:58 PM IDT)
• *David F.H. Dunkley (31/03/2019 04:42 AM IDT)
• *Oscar Volpatti (31/03/2019 09:17 AM IDT)
• *Yi Won Kim (31/03/2019 05:46 PM IDT)
• Radu-Alexandru Todor (31/03/2019 11:47 PM IDT)
• *Vladimir M. Milovanović (01/04/2019 01:49 AM IDT)