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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.

## March 2023 - Challenge

This riddle was suggested by Richard Gosiorovsky - Thanks, Richard!

A prime chain is a sequence p_1, p_2,\ldots, p_n of prime numbers such that every number in the chain is obtained from the previous one by adding one digit to the right. For example, the prime chain:
```7
73
739
7393
73939
739391
7393913
73939133
```
Is the longest prime chain (in base 10) where the first element has one digit. We say that p is a chained prime if it appears in a prime chain in base 10, starting from one-digit numbers. We can see that 73939133 is the largest chained prime. For any natural number n, we say that p is a n-exception chained prime if it appears in a prime chain where we allow up to n "exceptions" - elements of the chain that are not prime. For example, 3733799959397 is a 1-exception chained prime as can be seen from the chain:
```3
37
373
3733
37337
373379
3733799
37337999
373379995
3733799959
37337999593
373379995939
3733799959397
```
Note that p can also be non-prime, if it is considered one of the exceptions.

Your goal: Find the largest number which is a 5-exception chained prime.

A Bonus "*" will be given for finding the largest 5-exception reverse chained prime, where "reverse" means the elements of the chain are constructed by adding digits to the left.

For example, 29137 is a reverse chained prime, as can be seen by the chain:
```7
37
137
9137
29137
```

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: 01/03/2023 @ 11:50 AM EST
Solution: 25/04/2023 @ 11:00 AM EST
List Updated: 25/04/2023 @ 11:00 AM EST

#### People who answered correctly:

Lazar Ilic(1/3/2023 1:35 PM IDT)
Reiner Martin(1/3/2023 3:06 PM IDT)
*Alper Halbutogullari(1/3/2023 3:23 PM IDT)
Daniel Chong Jyh Tar(1/3/2023 6:00 PM IDT)
Andreas Puccio(1/3/2023 7:13 PM IDT)
Evan Schor(1/3/2023 10:01 PM IDT)
Todd Will(1/3/2023 10:59 PM IDT)
Michael Branicky(1/3/2023 11:44 PM IDT)
*K S(2/3/2023 6:34 AM IDT)
*Harald Bögeholz(2/3/2023 11:52 AM IDT)
Paul Revenant(2/3/2023 12:01 PM IDT)
Laurence Warne(2/3/2023 1:32 PM IDT)
*Bert Dobbelaere(3/3/2023 12:55 AM IDT)
*Gary M. Gerken(3/3/2023 5:05 AM IDT)
Michael Liepelt(3/3/2023 8:24 AM IDT)
Ethan (EJ) Watkins(3/3/2023 11:04 AM IDT)
Ralf Jonas(3/3/2023 12:14 PM IDT)
Kennedy(4/3/2023 3:48 PM IDT)
*Uoti Urpala(4/3/2023 5:02 PM IDT)
Clive Tong(4/3/2023 7:27 PM IDT)
*Vladimir Volevich(4/3/2023 7:32 PM IDT)
*Marco Bellocchi(5/3/2023 1:01 AM IDT)
Dieter Beckerle(5/3/2023 10:40 AM IDT)
*Latchezar Christov(5/3/2023 5:05 PM IDT)
Florian Sikora(5/3/2023 7:14 PM IDT)
*Sanandan Swaminathan(6/3/2023 11:08 AM IDT)
*Bertram Felgenhauer(6/3/2023 4:20 PM IDT)
Lorenzo Gianferrari Pini(6/3/2023 6:11 PM IDT)
*Edwin Hall(6/3/2023 8:47 PM IDT)
*Andreas Knüpfer(6/3/2023 10:21 PM IDT)
Dominik Reichl(7/3/2023 12:49 AM IDT)
*Tim Walters(7/3/2023 11:34 PM IDT)
Mark Rickert(10/3/2023 7:33 AM IDT)
Hakan Summakoğlu(10/3/2023 6:06 PM IDT)
*David Greer(10/3/2023 9:22 PM IDT)
Daniel Stanley(11/3/2023 1:04 AM IDT)
*Martin Thorne(11/3/2023 3:21 AM IDT)
*Peyo(11/3/2023 2:03 PM IDT)
*David F.H. Dunkley(11/3/2023 3:13 PM IDT)
*Chris Shannon(12/3/2023 8:43 PM IDT)
*Alexander Marschoun(13/3/2023 11:59 AM IDT)
*Stéphane Higueret(13/3/2023 12:58 PM IDT)
Luke Pebody(14/3/2023 11:39 AM IDT)
Anders Bisbjerg Madsen(15/3/2023 9:43 PM IDT)
Phil Proudman(16/3/2023 6:52 PM IDT)
Aaron Jackson(17/3/2023 12:03 AM IDT)
Sachal Mahajan(17/3/2023 1:15 AM IDT)
Kang Jin Cho(17/3/2023 5:33 PM IDT)
*Franciraldo Cavalcante(20/3/2023 3:07 AM IDT)
*Yan-wu He(20/3/2023 5:08 PM IDT)
Prashant Wankhede(20/3/2023 11:45 PM IDT)
Francisco Couzo(21/3/2023 8:53 PM IDT)
Daniel Bitin(21/3/2023 11:53 PM IDT)
Trigg Dudley(22/3/2023 11:59 AM IDT)
Andrew Wilson(22/3/2023 12:04 PM IDT)
Joshua Okonkwo(22/3/2023 12:06 PM IDT)
*Thomas Egense(24/3/2023 12:10 PM IDT)
*Stephen Herbert Jr(24/3/2023 3:49 PM IDT)
*José Eduardo Gaboardi de Carvalho(25/3/2023 9:04 AM IDT)
Lorenz Reichel(25/3/2023 5:55 PM IDT)
Reda Kebbaj(26/3/2023 10:54 AM IDT)
Virginia Ardévol Martínez(26/3/2023 3:59 PM IDT)
Wilder Boyden & Michael Leslie(26/3/2023 11:59 PM IDT)
Carl Löndahl(27/3/2023 11:27 AM IDT)
Mark Beyleveld(27/3/2023 6:00 PM IDT)
David Stevens(27/3/2023 10:12 PM IDT)
Victor Chang(28/3/2023 6:37 AM IDT)
*Nyles Heise(28/3/2023 6:38 AM IDT)
Dennie Tyshchenko(28/3/2023 3:04 PM IDT)
Davi Yan Macedo Brandão & Edson de Melo Neto(29/3/2023 3:14 AM IDT)
Shouky Dan & Tamir Ganor(29/3/2023 8:20 PM IDT)
Alfred Reich(30/3/2023 6:26 PM IDT)
*Yan-Wu He(30/3/2023 6:35 PM IDT)
Govind Raj Jujare(1/4/2023 8:46 PM IDT)
*Oscar Volpatti(1/4/2023 11:18 PM IDT)
Hansraj Nahata(1/4/2023 11:27 PM IDT)
Satya Tamby(2/4/2023 2:15 AM IDT)
Radu-Alexandru Todor(3/4/2023 2:41 AM IDT)
Motty Porat(3/4/2023 2:53 AM IDT)
Nathan Hopkins(5/4/2023 10:30 AM IDT)
Li Li(5/4/2023 7:57 PM IDT)
Evan Semet(17/4/2023 5:09 AM IDT)