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# Ponder This

Welcome to our monthly puzzles.
You are cordially invited to match wits with some of the best minds in IBM Research.

## June 2021 - Challenge

The numbers 32993 and 2097593 have the following curious properties:
1. They are both prime
2. They are both of the form $$a^b+b^a$$ for $$a,b > 1$$ integers
3. They are both of the form $$x^2+y^2$$ for $$x,y$$ integers
4. They are both of the form $$x^2+2y^2$$ for $$x,y$$ integers
5. They are both of the form $$x^2+7y^2$$ for $$x,y$$ integers

Where in equations 3-5 the x,y values can be different from equation to equation.

For example, for 32993 we have:
• $$2^{15}+15^2 = 32993$$
• $$143^2+112^2 = 32993$$
• $$15^2+2\cdot 128^2 = 32993$$
• $$25^2+7\cdot 68^2 = 32993$$

Your goal: Find another number satisfying those properties. Supply your answer in the following format: first the number itself, then the $$a,b$$ values that plug into the $$a^b+b^a$$ equation, and then the $$x,y$$ values solving equations 3-5.

For example, for 32993, the solution is given as:

32993
2 15
143 112
15 128
25 68


A bonus "*" will be given for finding a number greater than googol ($$10^{100}$$) that satisfies the properties.

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

Challenge: 31/05/2021 @ 12:00 PM EST
Solution: 04/07/2021 @ 12:00 PM EST
List Updated: 02/07/2021 @ 12:00 PM EST

*Uoti Urpala (31/5/2021 08:19 PM IDT)
*Bert Dobbelaere (31/5/2021 10:07 PM IDT)
*Yuping Luo (31/5/2021 11:54 PM IDT)
*Giorgos Kalogeropoulos (31/5/2021 11:57 PM IDT)
*Radu-Alexandru Todor (1/6/2021 1:33 AM IDT)
*Lazar Ilic (1/6/2021 5:41 AM IDT)
Sean Egan (1/6/2021 9:36 AM IDT)
*Reiner Martin (1/6/2021 10:41 AM IDT)
*Harald Bögeholz (1/6/2021 12:10 PM IDT)
*Paul Revenant (1/6/2021 1:18 PM IDT)
*Andreas Stiller (1/6/2021 3:51 PM IDT)
*Bertram Felgenhauer (1/6/2021 3:53 PM IDT)
*Latchezar Christov (1/6/2021 5:00 PM IDT)
*Albert Stadler (1/6/2021 9:21 PM IDT)
*Shouky Dan & Tamir Ganor (1/6/2021 11:04 PM IDT)
*Sean Egan (2/6/2021 12:16 AM IDT)
*Keith Schneider (2/6/2021 12:23 AM IDT)
*Hakan Summakoğlu (2/6/2021 12:41 AM IDT)
*Victor Chang (2/6/2021 7:12 AM IDT)
*Yash Pote (2/6/2021 8:48 AM IDT)
*Sanandan Swaminathan (2/6/2021 9:19 AM IDT)
*Daniel Chong Jyh Tar (2/6/2021 12:08 PM IDT)
*Alper Halbutogullari (2/6/2021 1:47 PM IDT)
*Michael Branicky (2/6/2021 1:53 PM IDT)
*Gary M. Gerken (2/6/2021 4:43 PM IDT)
*Kurt Foster (2/6/2021 4:44 PM IDT)
*Guillaume Escamocher (2/6/2021 7:09 PM IDT)
*Angus Chung (2/6/2021 9:50 PM IDT)
João Paulo Santos (3/6/2021 12:19 AM IDT)
*Jordan Rinder (3/6/2021 12:27 AM IDT)
*Yin Nam Dalton Fung (3/6/2021 4:00 AM IDT)
JJ Rabeyrin (3/6/2021 2:30 PM IDT)
*Dominik Reichl (3/6/2021 3:08 PM IDT)
*Hyeonseung Kwon (3/6/2021 3:54 PM IDT)
*Thomas Egense (3/6/2021 4:31 PM IDT)
Soonho You (3/6/2021 5:09 PM IDT)
*Hugh Taiyo Coleman (3/6/2021 7:56 PM IDT)
*Reda Kebbaj (3/6/2021 9:04 PM IDT)
Ahmet Yüksel(3/6/2021 9:06 PM IDT)
*Daniëlle van den Berg (3/6/2021 9:41 PM IDT)
*Dan Dima (3/6/2021 9:50 PM IDT)
*Tim Walters (4/6/2021 12:00 AM IDT)
*Chu-Wee Lim (4/6/2021 6:55 AM IDT)
Gürkan Koray Akpınar (4/6/2021 9:29 AM IDT)
*David Greer (4/6/2021 1:33 PM IDT)
*Clive Tong (4/6/2021 2:43 PM IDT)
*Daniel Scheinerman (4/6/2021 9:48 PM IDT)
*Maksym Voznyy (4/6/2021 10:49 PM IDT)
Steven Langerwerf(5/6/2021 12:36 AM IDT)
*Walter Schmidt (5/6/2021 2:50 PM IDT)
*Kang Jin Cho (5/6/2021 5:39 PM IDT)
Tung nguyen khac (7/6/2021 12:14 AM IDT)
Nir Drucker (7/6/2021 8:36 AM IDT)
Andrew Mullins (7/6/2021 10:37 PM IDT)
*Raymond Lo (8/6/2021 3:25 AM IDT)
*Michael Liepelt (8/6/2021 1:53 PM IDT)
*Stefan Wirth (9/6/2021 5:25 AM IDT)
Dieter Beckerle (9/6/2021 5:00 PM IDT)
*Kennedy (9/6/2021 6:28 PM IDT)
*Todd Will (9/6/2021 9:53 PM IDT)
*Thomas Pircher (10/6/2021 11:21 AM IDT)
Alexey Zyl (10/6/2021 12:27 PM IDT)
*Rainy Sokhonn (11/6/2021 9:03 PM IDT)
*Moning Zhang (12/6/2021 4:29 AM IDT)
Franciraldo Cavalcante (12/6/2021 5:37 PM IDT)
Lorenz Reichel (12/6/2021 8:33 PM IDT)
Fabio Michele Negroni (12/6/2021 8:38 PM IDT)
*Li Li (14/6/2021 7:10 AM IDT)
Joaquim Carrapa (14/6/2021 3:46 PM IDT)
*Marco Bellocchi (15/6/2021 12:58 PM IDT)
*Kai Guttmann (17/6/2021 12:37 AM IDT)
*Hansraj Nahata (17/6/2021 2:29 AM IDT)
*Govind Jujare (17/6/2021 3:50 AM IDT)
*Nyles Heise (18/6/2021 5:52 AM IDT)
Andreas Knüpfer (18/6/2021 4:20 PM IDT)
Chuck Carroll (19/6/2021 3:00 AM IDT)
*Jose Coelho (20/6/2021 12:09 AM IDT)
*Benjamin Phillabaum (20/6/2021 6:17 PM IDT)
Wang Longbu (21/6/2021 5:57 AM IDT)
*Guenter Nowak (21/6/2021 7:23 AM IDT)
*Matt Rips (21/6/2021 8:23 PM IDT)
*Soham Sud (22/6/2021 12:09 AM IDT)
*Sean Wellman (22/6/2021 7:12 PM IDT)
*Yusuf AttarBashi (22/6/2021 11:09 PM IDT)
*Chris Shannon (23/6/2021 6:24 AM IDT)
*Franciraldo Cavalcante (23/6/2021 4:08 PM IDT)
*Liubing Yu (24/6/2021 10:14 AM IDT)
*Hans Havermann (25/6/2021 7:18 AM IDT)
John Song (25/6/2021 10:02 AM IDT)
Arthur Vause (25/6/2021 4:39 PM IDT)
*Evert van Dijken (27/6/2021 7:39 PM IDT)
*Florian Fischer (28/6/2021 1:35 AM IDT)
*Dieter Beckerle (29/6/2021 12:36 PM IDT)
*Tomer Vromen (29/6/2021 11:15 PM IDT)
*Daniel Bitin (30/6/2021 1:02 AM IDT)
*Prashant Wankhede (30/6/2021 8:17 AM IDT)
*Tremaine Chimbari (30/6/2021 9:42 AM IDT)
Alex Fleischer (30/6/2021 10:24 AM IDT)
*Calvin Pozderac (30/6/2021 7:52 PM IDT)
*Alex Semenov (30/6/2021 8:37 PM IDT)
Eran Vered (30/6/2021 8:47 PM IDT)
*James Muir (1/7/2021 9:44 AM IDT)
*Oscar Volpatti (2/7/2021 7:17 AM IDT)
Armin Krauß (2/7/2021 3:33 PM IDT)
Motty Porat (2/7/2021 6:04 PM IDT)
*Yoishik Rakshit (2/7/2021 11:24 PM IDT)