Problems & Puzzles:
Puzzles
Puzzle 134.
The
1379CarrouselPrimes
Let's
ask for primes P that remains prime when the following set of
transformations applies at the same time:
a)
all the "1" digits transforms in "3" digits
b) all the "3" digits transforms in "7" digits
c) all the "7" digits transforms in "9" digits
d) all the "9" digits transforms in "1" digits
e) all the other digits (0, 2, 4, 5, 6 & 8), remain the same
('inactive' digits)
Let's
ask only for these primes P that remain prime under 3 successive
applications of the complete before mentioned set of transformations.
Here
are some easy examples:
I.
The
least carrousel prime with these properties (and at least on 'active'
digit) is: 19 > 31 > 73 >97. Incidentally
this carrousel prime has all its digits 'active' (to be honest the least
carrousel primes are 2 & 5)
II.
The
least carrousel palprime is: 131
> 373 > 797 > 919. Incidentally
this palprime has also all its digits 'active'.
III.
The
least carrousel prime that generates two couples of reversible primes
is:157 > 359
> 751 > 953
(unfortunately it has one 'inactive' digit)
IV.
The
least carrousel prime with only one digit 'active' is:
821 > 823
> 827 > 829
V.
The least carrousel prime with only one 'inactive' digits is:
2111>2333>2777>2999
Questions:
1.
Find 3 more carrousel palprimes,
having
only 'active' digits.
2.
Find 3 carrousel primes that generates, each, two couples of reversible
primes, having only 'active' digits.
3.
Find one Titanic
(*) carrousel prime having only one 'active' digit.
4.
Find one 100 digits
carrousel prime having only one 'inactive' digit.
______
(*) As a fair & instructive beginning I
can accept smaller solutions if they are composed
of 100, 200, ..., etc. digits.
As maybe you have noticed these asked four primes may form two couples of twins in the
same decade.
Solution
Michael Bell has produced solutions to questions
1 & 3. Here is his email (22/4/01):
"I've found 2 more carrousel palprimes with only
active digits:
3193391713171933913 > 7317713937393177137 >
9739937179717399379 > 1971179391939711791 and
7319171773771719137 > 9731393997993931379 >
1973717119117173791 > 3197939331339397913
Also, as a first try at question 3 I found the quadruplet
46060600066600606006*10^30 + 1,3,7,9. Which has 50 digits and only 1
active digit."
Michael is not sure that the palprime gotten is
the earliest. He also think that to get a titanic solution to question 3
is "close to impossible, unless someone can see a way of sieving
efficiently "
***
Giuliano Daddario has solved (2/9/01) the
question 2, founding 5 examples with the asked properties:
{139397, 371719, 793931, 917173}
{193937, 317179, 739391, 971713}
{1173193939379177,3397317171791399,7719739393913711,9931971717137933}
{113799319379331977, 337911731791773199, 779133973913997311,
991377197137119733}
{139717391739171397, 371939713971393719,
793171937193717931, 917393179317939173}
***
