Problems & Puzzles: Puzzles

Puzzle 17.- Weakly Primes

Following the problem 12 of the "South Missouri State University archive of puzzles" ( http://math.smsu.edu/~les/POW12.html ), a weakly prime is any prime that lost his primality condition by changing - one at a time-anyone of it’s digits to any number (0-9) other than the current.

In base 10 the first 3 weakly primes are : 294001, 505447, 584141.

I’ll keep here the first one and the 10 largest weakly primes base 10 :

 Weakly Primes base 10 The first 294001(Obtained by Ken Duisenberg) The 10 largest 18 999999999997802311 J. McCranie (17/08/98) 18 999999999998270057 "" 18 999999999998832431 "" 21 999999999999999543767 Robert T. McQuaid 50 (9)44 649691 C. Rivera 61 1(0)53 2236743 C. Rivera 81 1(0)73 1295823 C. Rivera 101 1(0)94 590181 C. Rivera 151 1(0)144 366303 C. Rivera 201 1 (0)193 3592453 C. Rivera 251 1 (0)243 1856301 Tiziano Mosconi 23/8/01

On March 2007, J. K. Andersen wrote:

A 1000-digit weakly prime: (17*10^1000-17)/99+21686652 = (17)_496 38858369. Found with PrimeForm/GW and proved with Marcel Martin's Primo.

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On Jan 26, 2021 Dana Jacobsen wrote:

I know this one is quite old, but after a few recent papers on Arxiv about delicate primes, I added the simple function to my library.  Some searching afterwards found this puzzle.

In celebration of 2021:

#  500 digits: 2021 * 10^(500-4) + 7543997
# 1000 digits: 2021 * 10^(1000-4) + 2550219
# 2021 digits: 2021 * 10^(2021-4) + 4523733

Found using Perl/ntheory.

Double checked prime with Pari 2.14 ECPP, and double checked as a delicate/weakly prime using Michel Marcus' Pari function.

[delicate primes or weakly primes?]

I believe they are the same (technically "digitally delicate primes" to reference base 10).  OEIS A050249, "Weakly prime numbers … Also called digitally delicate primes."

The following is just trivia because it was interesting for me to search.  There is no consistent answer.

Klamkin's 1978 question doesn't seem to call them anything, nor does Tao's paper.
I'm not able to track down Erdos' 1979 "Solution to Problem 1029" to see what he says.
Mathworld and Anderson's Wikipedia article calls them Weakly Primes, referencing your page.
Hopper and Pollack's 2015 arxiv paper calls them "digitally delicate":

"The infinitude of the primes appearing in Corollary 1.2 had earlier been shown by Erd˝os [2].
(He assumes a = 10 but the argument generalizes in an obvious way.)
When a = 10, these “digitally delicate” primes are tabulated as sequence A050249 in the OEIS, where they are called “weakly prime”.

The papers by Filaseta and Southwick, most in the last decade, also call them digitally delicate.

Emily Stamm's 2020 preprint calls them weakly prime, and references most of the other papers so clearly the author is familiar with both terms and chose that one.

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