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 timeanyone of it’s digits to any number (09) 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 1000digit weakly prime:
(17*10^100017)/99+21686652 = (17)_496 38858369. Found with
PrimeForm/GW and proved with Marcel Martin's Primo.
***
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^(5004) + 7543997
# 1000 digits: 2021 * 10^(10004) + 2550219
# 2021 digits: 2021 * 10^(20214) + 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.
***
