Ponder This Challenge - June 2019 - Two close sums of the square root of smooth numbers

Let's define a *smooth* number as a natural number with prime factors that are single-digit (less than 10).
An example of such a number is 1560674304, which is related (how?) to the 108th birthday of IBM.

This month's challenge is to find two sets of smooth numbers whose square root sums are close to each other.
For example, the sets $\{2,6\}$ and $\{15\}$ yield a distance between their square root sums of less than 0.01 since $\sqrt{2}+\sqrt{6} = 3.86 \ldots \approx 3.87\ldots = \sqrt{15}$.

Two sets that meet our criterion even better are $\{2,10,15\}$ and $\{6,36\}$, which yield an even smaller distance between their square root sums: $\sqrt{2}+\sqrt{10}+\sqrt{15} = 8.44947\ldots \approx 8.44948\ldots = \sqrt{6}+\sqrt{36}$.

Find two sets of smooth numbers that produce a distance d where $0 < d < 10^{-15}$.

Bonus question: Find a smooth number that is connected to last month's bonus question (a date in May 2019).

Solution

• Click here to view the solution

  There are small solutions, like Johan Van Mengsel's solution: [20, 72, 75, 105, 120 ,162 ,336], [2, 7 ,14 ,40 ,42 ,96 ,126, 150, 400]

  Reiner Martin was the first to solve it using the rational approximation of sqrt(2). You can read about it on Thomas Huet's blog https://thomash.fr/2019/06/15/ibm-ponder-this-june-2019.html.
  Jingran Lin had a nice solution:
  Rewrite the example given on the website: $6 + \sqrt{6} - \sqrt{2} - \sqrt{10} - \sqrt{15} < 1.52 \cdot 10^{-5}$
  Hence:
  $(6 + \sqrt{6} - \sqrt{2} - \sqrt{10} - \sqrt{15}) ^4 < 5.31\cdot 10^{-20}$
  Expand to give:
  (*)

  Rewrite the following to sum/diff of smooth numbers, if not already:
  15281= 15309 - 28
  3864= 3888 - 24
  2968= 3000 - 32
  2328 = 2400 - 72
  6092 = 6272 - 180
  4740 = 4800 - 60
  3920 smooth number
  980 smooth number

  Expand (*) further:

  Take the square of each element in the above to produce two sets of smooth numbers:
  {234365481, 1152, 3072,28800000, 236027904, 36000, 28812000}
  {784, 30233088, 27000000, 25920, 194400, 230400000, 230496000}

  As for the bonus question, $1560674304 = 2^{17} \cdot 3^{5} \cdot 7^{2}$ is the number of seconds elapsed between the UNIX Epoch time (00:00:00 Thursday, 1 January 1970) and the 108th birthday of IBM (Sunday, June 16, 2019).
  We were aiming for a similar smooth number, $612615150 = 2\cdot 3^{6}\cdot 5^{2}\cdot 7^{5}$, which corresponds to (Wednesday, May 31, 1989), the 30th anniversary of IBM Academy of Technology.
  But we accepted other solutions like the one submitted by Chris Shannon, who found that the smooth number 1210104000, interpreted as a UNIX time stamp, results in May 6, 2008, the date when Fran Allen received the Erna Hamburger Distinguished Lecture Award at EPFL.
  Others found that the year 2016 is smooth ($2016=2^{6}\cdot 3^{2}\cdot 7$) and the last woman to become president of the IBM Academy of Technology was Susan Schreitmueller Smartt in 2016).
  and other solutions discovered that 2000 is smooth as well ($2000 = 2^{4}\cdot 5^{3}$), when Frances E. Allen became a fellow of the Computer History Museum.

Solvers

• Oleg Vlasii (29/05/2019 07:43 PM IDT)
• Uoti Urpala (29/05/2019 08:50 PM IDT)
• *Bert Dobbelaere (29/05/2019 10:57 PM IDT)
• Reiner Martin (30/05/2019 11:45 AM IDT)
• Thomas Huet (30/05/2019 02:01 PM IDT)
• *Alper Halbutogullari (30/05/2019 08:55 PM IDT)
• Lorenz Reichel (30/05/2019 09:33 PM IDT)
• Jingran Lin (31/05/2019 12:48 AM IDT)
• Anders Kaseorg (31/05/2019 04:49 AM IDT)
• *Eden Saig (01/06/2019 02:48 AM IDT)
• *Florian Fischer (01/06/2019 02:19 PM IDT)
• Joseph DeVincentis (01/06/2019 07:32 PM IDT)
• Andreas Stiller (02/06/2019 03:08 AM IDT)
• Guenter Nowak (02/06/2019 10:34 AM IDT)
• Boris Silantev (02/06/2019 03:16 PM IDT)
• Alex Fleischer (03/06/2019 02:40 PM IDT)
• Richard Gosiorovsky (03/06/2019 05:21 PM IDT)
• Liubing Yu (04/06/2019 06:18 AM IDT)
• Harald Bögeholz (04/06/2019 04:28 PM IDT)
• John Tromp (04/06/2019 04:29 PM IDT)
• Deron Stewart (04/06/2019 07:04 PM IDT)
• Todd Will (04/06/2019 07:15 PM IDT)
• *Haye Chan (05/06/2019 10:00 AM IDT)
• George Wahid (05/06/2019 05:38 PM IDT)
• Michael van Fondern (05/06/2019 08:38 PM IDT)
• Tom Chen (06/06/2019 07:06 AM IDT)
• Albert Stadler (07/06/2019 12:02 AM IDT)
• Lachezar Hristov (07/06/2019 03:01 PM IDT)
• *Kang Jin Cho (07/06/2019 06:46 PM IDT)
• Klaus Hoffmann (07/06/2019 07:28 PM IDT)
• Vladimir Volevich (07/06/2019 10:33 PM IDT)
• Rashid Naimi (08/06/2019 04:01 AM IDT)
• JJ Rabeyrin (08/06/2019 02:35 PM IDT)
• Jacoby Jaeger (09/06/2019 01:27 AM IDT)
• Noam Lasry (09/06/2019 05:36 PM IDT)
• Xiao Liu (10/06/2019 09:02 AM IDT)
• Johan Van Mengsel (10/06/2019 09:18 AM IDT)
• Loukas Sidiropoulos (10/06/2019 11:53 PM IDT)
• *Daniel Chong Jyh Tar (11/06/2019 10:31 AM IDT)
• Yasodhar Patnaik (11/06/2019 06:53 PM IDT)
• Robert Lang (12/06/2019 02:24 PM IDT)
• Jan Fricke (12/06/2019 10:46 PM IDT)
• Amos Guler (13/06/2019 10:27 AM IDT)
• Aaron Zolnai-Lucas (13/06/2019 06:51 PM IDT)
• Dieter Beckerle (13/06/2019 11:44 PM IDT)
• Daniel Bitin (14/06/2019 10:41 AM IDT)
• Hermann Jurksch & Hugo Pfoertner (15/06/2019 01:54 AM IDT)
• Pål Hermunn Johansen (16/06/2019 02:04 PM IDT)
• *Miguel Sauam & Rogerio Ponce da Silva (21/06/2019 06:25 AM IDT)
• *Chris Shannon (21/06/2019 12:11 PM IDT)
• Gareth Ma (22/06/2019 11:39 AM IDT)
• Charles Joscelyne (23/06/2019 10:04 PM IDT)
• David Greer (24/06/2019 02:43 PM IDT)
• Peter Gerritson (24/06/2019 10:32 PM IDT)
• Kevin Naslund (25/06/2019 08:17 AM IDT)
• Jimmy Waters (26/06/2019 07:08 PM IDT)
• Muralidhar Seshadri (27/06/2019 09:53 AM IDT)
• Calvin Pozderac (27/06/2019 05:08 PM IDT)
• Chen Zixuan (29/06/2019 07:59 PM IDT)
• Motty Porat (30/06/2019 05:53 PM IDT)
• Radu-Alexandru Todor (01/07/2019 02:04 AM IDT)
• Marco Bellocchi (02/07/2019 02:51 AM IDT)
• *Li Li (02/07/2019 07:24 AM IDT)
• Nyles Heise (02/07/2019 08:11 AM IDT)
• Oscar Volpatti (03/07/2019 10:56 AM IDT)