Problems & Puzzles:
Puzzles
Puzzle 147.
Twin & reversible primes
The first non trivial example of twin & reversible
primes is:
(71, 73) & (17, 37)
A somewhat larger example is:
(173313197,
173313199) &
(791313371, 991313371)
One more example I have calculated for this puzzle is:
(10^60+68599471,
10^60+68599473) & (17499586*10^53+1,
37499586*10^53+1)
Question:
Calculate
a) a semi-titanic
couple b) a titanic
couple
Jim Fougeron found (11/1/2002) larger examples of
the asked pairs, still not a semi-titanic (500 digits) but large enough to
make a record by the moment:
Sub titanic twin/reversible pair of primes:
244909752*809#-1 & 244909752*809#+1 and their reverses. These
are
all 341 digits in length. They have been proven. The
primorials
using N+1 and N-1 proving techniques, and the reverses by using
ECPP with Primo.
Here are the expanded forms:
244909752*809#-1
928062018516027919593015654952392007884049452133361393716590
124875109935157427807098311906265022073108514282881621049445
449102273154312271007547102727569493270274380318853077771489
952353573686998025706149170380122336228389501475680847773612
187258820238957154928292297037244068743909043939206586014530
51834078065146621961586535712689123624079
244909752*809#+1
928062018516027919593015654952392007884049452133361393716590
124875109935157427807098311906265022073108514282881621049445
449102273154312271007547102727569493270274380318853077771489
952353573686998025706149170380122336228389501475680847773612
187258820238957154928292297037244068743909043939206586014530
51834078065146621961586535712689123624081
Rev(244909752*809#-1)
970426321986217535685169126641560870438150354106856029393409
093478604427307922928294517598320288527812163777480865741059
838226332210830719416075208996863753532599841777703588130834
720723949657272017457001722134513722019445449401261882824158
013702205626091138907087247515399015784210956173931633312549
40488700293259456510395919720615810260829
Rev(244909752*809#+1)
180426321986217535685169126641560870438150354106856029393409
093478604427307922928294517598320288527812163777480865741059
838226332210830719416075208996863753532599841777703588130834
720723949657272017457001722134513722019445449401261882824158
013702205626091138907087247515399015784210956173931633312549
40488700293259456510395919720615810260829
***
J. K. Andersen wrote:
The smallest case with semi-titanic twins is
(10^499+174295123051, 10^499+174295123053) &
(150321592471*10^488+1, 350321592471*10^488+1).
The GMP library made prp tests. Primo proved the primes.
***
|
