Six prime chunks

(Dall-e creation)

 

Here are, just for fun, the first 123 pandigital numbers (the first 1000 ones are listed here):

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, 1023457968, 1023457986, 1023458679, 1023458697, 1023458769, 1023458796, 1023458967, 1023458976, 1023459678, 1023459687, 1023459768, 1023459786, 1023459867, 1023459876, 1023465789, 1023465798, 1023465879, 1023465897, 1023465978, 1023465987, 1023467589, 1023467598, 1023467859, 1023467895, 1023467958, 1023467985, 1023468579, 1023468597, 1023468759, 1023468795, 1023468957, 1023468975, 1023469578, 1023469587, 1023469758, 1023469785, 1023469857, 1023469875, 1023475689, 1023475698, 1023475869, 1023475896, 1023475968, 1023475986, 1023476589, 1023476598, 1023476859, 1023476895, 1023476958, 1023476985, 1023478569, 1023478596, 1023478659, 1023478695, 1023478956, 1023478965, 1023479568, 1023479586, 1023479658, 1023479685, 1023479856, 1023479865, 1023485679, 1023485697, 1023485769, 1023485796, 1023485967, 1023485976, 1023486579, 1023486597, 1023486759, 1023486795, 1023486957, 1023486975, 1023487569, 1023487596, 1023487659, 1023487695, 1023487956, 1023487965, 1023489567, 1023489576, 1023489657, 1023489675, 1023489756, 1023489765, 1023495678, 1023495687, 1023495768, 1023495786, 1023495867, 1023495876, 1023496578, 1023496587, 1023496758, 1023496785, 1023496857, 1023496875, 1023497568, 1023497586, 1023497658, 1023497685, 1023497856, 1023497865, 1023498567, 1023498576, 1023498657, 1023498675, 1023498756, 1023498765, 1023546789, 1023546798, 1023546879.

The integer 2340156789 is also a pandigital number.

Which can be cut into 6 prime-chunks: _2_3_401_5_67_89_

Question #1

Are there other such pandigitals?

Swapping 4 and 6 from the above one, I found 2360154789:  _2_3_601_5_47_89_ 

I guess 7 prime-chunks are impossible.

___________________________

Next day update

Giorgos Kalogeropoulos was quick to send this:

> There exist 9360 pandigital numbers that can be cut into 6 prime-chunks.

> We can find them from all the permutations (720 for each set) of the following sets:

{2, 3, 5, 67, 89, 401}, 

{2, 5, 7, 61, 83, 409}, 

{2, 5, 7, 43, 89, 601}, 

{2, 3, 5, 47, 89, 601}, 

{2, 3, 5, 41, 89, 607}, 

{2, 3, 5, 41, 67, 809}, 

{2, 5, 7, 43, 61, 809}, 

{2, 3, 5, 47, 61, 809}, 

{2, 3, 5, 7, 461, 809}, 

{2, 3, 5, 7, 641, 809}, 

{2, 3, 5, 7, 41, 6089}, 

{2, 3, 5, 7, 41, 8069}, 

{2, 3, 5, 7, 41, 8609}.

___________________________

EA (follow up of the original post)

... No pandital number is prime, of course, as the sum 0+1+2+3+4+5+6+7+8+9=45.

If digits may appear multiple times, pandigitals primes are easy to find, see here.

And for the fun again, the first 123 such pandigital primes are:

10123457689, 10123465789, 10123465897, 10123485679, 10123485769, 10123496857, 10123547869, 10123548679, 10123568947, 10123578649, 10123586947, 10123598467, 10123654789, 10123684759, 10123685749, 10123694857, 10123746859, 10123784569, 10123846597, 10123849657, 10123854679, 10123876549, 10123945687, 10123956487, 10123965847, 10123984657, 10124356789, 10124358697, 10124365879, 10124365987, 10124369587, 10124378569, 10124385967, 10124389567, 10124395867, 10124398657, 10124536789, 10124538769, 10124563789, 10124563879, 10124563987, 10124568793, 10124576893, 10124578693, 10124579863, 10124583967, 10124586397, 10124589637, 10124593867, 10124596873, 10124597683, 10124635879, 10124635897, 10124638759, 10124659873, 10124673859, 10124678953, 10124683759, 10124685379, 10124687359, 10124693857, 10124695783, 10124695837, 10124735689, 10124736859, 10124758639, 10124759863, 10124763589, 10124769583, 10124785369, 10124785639, 10124795683, 10124798653, 10124835697, 10124835769, 10124836759, 10124853679, 10124859673, 10124865739, 10124865973, 10124867359, 10124867539, 10124867593, 10124869357, 10124873659, 10124876539, 10124895763, 10124896357, 10124935687, 10124957683, 10124958673, 10124963587, 10124965387, 10124965783, 10124968537, 10124968753, 10124983567, 10124987563, 10125348769, 10125364897, 10125367849, 10125368749, 10125374869, 10125396847, 10125436879, 10125436987, 10125463897, 10125463987, 10125468739, 10125468973, 10125473689, 10125476839, 10125479863, 10125483769, 10125483967, 10125493687, 10125496783, 10125638947, 10125648937, 10125674983, 10125683479, 10125684739, 10125693847.

Question #2 

[a] What are the smallest pandigital primes that can be cut into 7 prime-chunks?

[b] 8 prime chunks? 

[c] 9 prime chunks?

____________________

Update #2 by GK

> [a] There are 4397 pandigital primes that can be cut into 7 prime-chunks. Here are the first ten ones:

 _2_3_401_5_89_67_7_

 _2_3_401_67_5_7_89_

 _2_3_401_67_5_89_7_

 _2_3_401_7_5_67_89_

 _2_3_401_7_67_5_89_

 _2_3_401_89_67_5_7_

 _2_3_41_5_7_809_61_

 _2_3_41_5_809_7_61_

 _2_3_41_5_89_601_7_

 _2_3_41_5_89_7_601_

> [b] The pandigital primes that can be cut into 8 prime-chunks must have at least 13 digits. There are in total 875653 such 13-digit primes. The first ten ones are:

_103_2_3_41_5_89_67_7_

_103_2_3_41_7_5_89_67_

_103_2_3_43_67_89_5_7_

_103_2_3_43_7_5_89_67_

_103_2_3_43_7_67_5_89_

_103_2_3_47_61_7_5_89_

_103_2_3_47_67_5_89_7_

_103_2_3_47_7_89_5_67_

_103_2_3_47_89_61_5_7_

_103_2_3_5_41_61_89_7_

> There are 335 different chunks that build these numbers. Here are the first 10:

{2, 3, 5, 7, 41, 61, 83, 109} {2, 3, 5, 7, 41, 61, 89, 103} {2, 3, 5, 7, 43, 61, 83, 109} {2, 3, 5, 7, 41, 61, 89, 107}

{2, 3, 5, 7, 43, 61, 89, 103} {2, 3, 5, 7, 41, 61, 89, 109} {2, 3, 5, 7, 41, 67, 83, 109} {2, 3, 5, 7, 43, 61, 89, 107} {2, 3, 5, 7, 41, 67, 89, 103} {2, 3, 5, 7, 43, 61, 89, 109}

> [c] The pandigital primes that can be cut into 9 prime-chunks must have at least 15 digits.

There are 3764 sets that produce 69061759 primes with 15 digits! The first one is 103132343589761 –> {103, 13, 2, 3, 43, 5, 89, 7, 61} First few primes: 103132343589761, 103132343675897, 103132343761589, 103132343789561, 103132343895677, 103132343895767, 103132357614389, 103132357674389, 103132357894361, 103132361743589, 103132367743589, 103132367754389, 103132367894357, 103132367895743, 103132374358961, 103132375674389, 103132375894367, 103132376189543, 103132378954361, 103132378967543, 103132389576743, 103132389675743, 103132433756189, 103132433756789, 103132433895677, 103132433895767, 103132433897561, 103132435361789, 103132435361897, 103132435389677, ...

What a great job, Giorgos! The [a], [b] and [c], sequences were submitted to the OEIS this December 9, 2023. Many thanks!

_______________________

December 20th update – Gilles Esposito-Farèse

(in French, but easy to understand)

Voici le plus petit nombre premier pandigital qui est la concaténation des chiffres de deux nombres eux-mêmes premiers : 101, 23465789 ou 101234657, 89 [Les suivants sont 101, 23654789, puis 101, 23956487, puis 101, 23965847, et j'ai fait le même calcul pour les 1000 premiers pandigitaux de l'OEIS.] Si le but est d'obtenir le plus petit nombre premier parmi les morceaux, alors on trouve 1012486597, 3. [On ne peut pas obtenir 2 sinon le nombre concaténé serait pair, en tout cas dans cette liste de l'OEIS. Faisable avec de grands nombres premiers commençant par 2, bien sûr.] En décomposant en trois morceaux, je trouve 101, 2384659, 7 puis 101, 2456879, 3 puis 101, 2457869, 3 puis 101, 245863, 97 ... Le premier morceau "2" est obtenu dans 101435869, 2, 7. En décomposant en quatre morceaux : 101, 23, 4657, 89 puis 101, 2, 346589, 7 (où il y a déjà un morceau "2") ... En décomposant en cinq morceaux : 101, 2, 3, 4657, 89 puis 101, 2, 3, 46589, 7 ... En décomposant en six morceaux : 101, 2, 43, 5, 67, 89 puis 101, 2, 43, 89, 5, 67 puis 101, 2, 463, 5, 89, 7 puis 101, 2, 467, 89, 5, 3 ... Et tu as déjà les résultats pour sept morceaux [Update #2 by GK]. ____________________ Tous ces calculs sont évidemment faisables avec des nombres hétéropandigitaux (= exactement les dix chiffres, qui ne sont donc pas eux-mêmes premiers). Pour deux morceaux : 102345689, 7 puis 102345869, 7 puis 10234589, 67 puis 1023467, 859 ... Il n'y a pas de morceaux "2" ni "3" dans les 1000 premiers hétéropandigitaux de l'OEIS. Pour trois morceaux : 1023467, 5, 89 puis 1023467, 89, 5 ... Le premier morceau "3" est dans 1024589, 3, 67 et il n'y a pas de morceau "2" dans les 1000 premiers hétéropandigitaux de l'OEIS. Pour quatre morceaux : 10243, 5, 67, 89 et ses cinq autres permutations (par exemple 10243, 89, 67, 5) Et pour les 1000 premiers hétéropandigitaux de l'OEIS, aucune décomposition en cinq morceaux (ni six, ni sept) n'est possible (avec des morceaux premiers).

Merci beaucoup Gilles ! Beau travail !

_______________________

EA (follow up of the original post)

Writing this page I bumped into this – which was fun to re-read after almost 14 years!

http://www.cetteadressecomportecinquantesignes.com/DixChiffres.htm

(Dall-e creation)