Ponder This Challenge - March 2023 - Chained Primes

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:

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:

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:

Solution

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  March 2023 Solution:

  The solutions are

  3733799911799539139382193991 (chained prime)

  996381323136248319687995918918997653319693967 (reversed chained prime)

  An implicit assumption, pointed out by many solvers, was that when considering reversed chained primes, we must disallow leading zeros, meaning that zero cannot appear anywhere in the number except as the least significant digit. In the words of Aaron Jackson (thanks, Aaron!):

  “In regards to the bonus, I believe it to be a trick question. Finding the largest 5-exception reverse chained prime, let alone the largest 0-exception reverse chained prime, is (practically) impossible. Considering there exists an infinite number of prime numbers, it is possible a prime number exists consisting of some number followed by perhaps an infinite number of 0's and ending with a 1, 3, or 7 (or, given we can use skips, any number) that could be prime.”

  This gives rise to a theoretical question (perhaps a simple one): Given some reversed chained prime p, are there infinite primes of the form “1000...000p"?

Solvers

• 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)