Problems & Puzzles:
Puzzles
Puzzle 72. Persistent Palprimes
Patrick De Geest in his alwaysinteresting
pages
asked for the primality condition of a palprime after a digit d
(from 0 to 9) is inserted between its adjacent digits.
De Geest found that:
"13331 is
the smallest palprime with the following property: Inserting any
digit d between adjacent digits of this palprime never produces a new
prime !".
Later he also found that 131
remains prime 6 times out of 10:
10301
= prime
11311
= prime
12321
= 3 x 3 x 37 x 37
13331
= prime
14341
= prime
15351
= 3 x 7 x 17 x 43
16361
= prime
17371
= 29 x 599
18381
= 3 x 11 x 557
19391
= prime
I have found that 7762868682677
remains prime 7 times out of 10
(for d = 0, 2,
4, 5, 6, 7, 8)
Here is the latest state of affairs (24/10/99) concerning the smallest
existing Persistent Palprimes for the following
eleven possible cases :
0 out of 10 = 13331
1 out of 10 = 101
2 out of 10 = 383
3 out of 10 = 151
4 out of 10 = 11311
5 out of 10 = 353
6 out of 10 = 131
7 out of 10 = 7762868682677
8 out of 10 = ?
9 out of 10 = ?
10 out of 10 = ?
With Patrick's kind permission I can now bring his puzzle to you
through
these PP&Ppages and added the following extra questions :
1. Is it theoretically possible that a palprime  modified according to the
above prescribed insertion rule 
remains prime 8, 9 or 10 times out of 10?
2.Can you find betterranked palprimes
than 7762868682677
for the 7 out of 10 or is this the
smallest possible palprime? 3.Can
you find the smallest palprime for the remaining cases '8 out of
10', '9 out of 10' and '10 out of 10'.
4. Can you redo the exercise but this
time with 'composites' and 'primes' instead of only 'palprimes' ?
Felice Russo sent (26/11/99) his results for the item 4.
of this puzzle: "For prime numbers I found:
n
#p

439
0
101
1
31
2
29
3
53
4
11
5
17
6
1933
7
1411789 8
where n is the smallest persistent prime number and #p the number of
primes generated. No solution for #p=9 and 10 up to 78*10^6.
For composite numbers instead I obtained:
n #p

121 0
111 1
69 2
27 3
33 4
21 5
49 6
1513 7
5809 8
and no solution for #p=9 and 10 up to 8.2*10^6."
***
Felice Russo (20/12/99) "Item 2. No solution
for rank 7/10 has been found up to 10^11." Later
(10/1/2000) he added: "The palprime 7762868682677 is the
smallest one for the rank 7/10."
*** On March 2011, Giovanni Resta wrote:
I studied puzzle 72 a little. These are my
findings:
Question Q3 (palprimes)
the smallest prime palindrome which produces 8 out of 10
primes is 7792235971795322977
(This was not asked, but the smallest palindrome
composite for 7/10 is 1190906090911 and
for 8/10 is 9605826109016285069).
If there are palindromes (prime or composite)
which produces 9 or 10 primes, they must have
at least 25 digits.
Question Q4 (non palindromes)
For non palindromes we have,
for composites:
9: 1048435661
10: 7691546233
and for primes:
9: 1485444967
10: 1003229774283941
This last value it is also the missing
value for the solution of Puzzle 557 (Q1).
This last finding was a lucky one, since it is quite
at the beginning of the set of numbers with 16 digits.
I examined all the numbers up to 13 digits in
3 or 4 days thanks to a little modular trick
that let me speed up the search about 100 times.
And it is easy to see that there can not be
solutions with 10/10 for numbers with 14 or 15 digits, so
the next step was to search 16digit numbers.
***
