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Bonjour à tous,
3 en décimal est premier et son écriture binaire 11 est aussi premier.
Ce cas est assez rare malgré la présence des 1.
Sauriez vous donner la plus grande liste de tels cas?
Bonjour Sylvieg
C'est un bon début j'en suis à 331 .dans les binaires à 9 chiffres je stoppe pour le moment 
Bonjour,
(3, 11),
(5, 101),
(23, 10111),
(47, 101111),
(89, 1011001),
(101, 1100101),
(149, 10010101),
(157, 10011101),
(163, 10100011),
(173, 10101101),
(179, 10110011),
(199, 11000111),
(229, 11100101),
(313, 100111001),
(331, 101001011),
(367, 101101111),
(379, 101111011),
(383, 101111111),
(443, 110111011),
(457, 111001001),
Quelques cas additionnels...
Cliquez pour afficher(3, 11)
(5, 101)
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(1048571, 11111111111111111011)
>mdr_non
Je pense que le sujet est clos 
As-tu un générateur de nombre premiers ?
Pour la modération
Je pense qu'il suffit de donner les 10 premiers et les 10 derniers de la liste
* Sylvieg edit> *
Cliquez pour afficher(3, 11)
(5, 101)
(23, 10111)
(47, 101111)
(89, 1011001)
(101, 1100101)
(149, 10010101)
(157, 10011101)
(163, 10100011)
(173, 10101101)
??
(1046791, 11111111100100000111)
(1046977, 11111111100111000001)
(1046999, 11111111100111010111)
(1047199, 11111111101010011111)
(1047313, 11111111101100010001)
(1047689, 11111111110010001001)
(1047833, 11111111110100011001)
(1048063, 11111111110111111111)
(1048343, 11111111111100010111)
(1048571, 11111111111111111011)
(5, 101)
(23, 10111)
(47, 101111)
(89, 1011001)
(101, 1100101)
(149, 10010101)
(157, 10011101)
(163, 10100011)
(173, 10101101)
??
(1046791, 11111111100100000111)
(1046977, 11111111100111000001)
(1046999, 11111111100111010111)
(1047199, 11111111101010011111)
(1047313, 11111111101100010001)
(1047689, 11111111110010001001)
(1047833, 11111111110100011001)
(1048063, 11111111110111111111)
(1048343, 11111111111100010111)
(1048571, 11111111111111111011)
Bonjour
Une variante qui semble monstrueuse :
1°) On part d'un entier premier par exemple 5 .
2°) On donne son écriture binaire et on lit le résultat comme une écriture décimale . Par exemple 5 donne 101 . Si on obtient un nombre premier on continue ici 101 est premier .
3°) On donne l'écriture binaire de 101 et on lit le résultat comme une écriture décimale . L'écriture binaire de 101 est 1100101 et 1100101 est premier : on continue .
4°) ...
Ici on obtient : 5 -> 101 -> 1100101 -> ? c'est à dire une chaîne de longueur au moins 3 . On peut sûrement faire plus long mais les calculs deviennent vite horribles 
Imod
>Imod
Excellente évolution .
Aucun problème pour créer un binaire ,mais pour générer une liste de premiers ...
En son temps Jamo avait donné une liste jusqu'à 1 000 000 que j'ai fait évoluer jusqu'à 2 000 000 mais quid au delà
Oui même avec un robot ça semble monstrueux mais avec deux bases plus proches par exemple 9 et 10 , ça peut être amusant à condition de choisir le nombre de départ plus grand que la petite base sinon la suite est infinie , constante et sans intérêt .
Imod
bonjour,
pour l'itération il n'y a pas besoin de table mais de déterminer si un nombre (grand) est premier ou pas
1 100101 est bien premier, il s'écrit en binaire 100001100100101000101 qui n'est pas premier (multiple de 7)
donc la chaine commencée à 5 par Imod s'arrête là.
(outils utilisés : conversion par la calculatrice de Windows, test de primarité et factorisation par le site 
Alpertron)
Tout à fait, on peut utiliser un test de primalité (Miller-Rabin, ...).
Voici les premières chaînes record:
[2]
[3, 11]
[5, 101, 1100101]
[3893257, 1110110110100000001001, 1111000010110111011111010111111000010101000000010001101100101111101001, 10100100110101100100110110101110101001011100101111010101001000011110010110001100110011101010101000011011001110111111001111110110101000101101100000100001011011100101011111100010000011101101111011111100110111000111111011011001001001]Il n'y a pas de chaîne de longueur 5 pour un nombre de départ inférieur à 150 000 000.
AnnulerRépondre à ce sujet
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Cours et Exercices de maths
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Outils de maths
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Pages les plus populaires
- Théorême de Thalès et réciproque
- Trigonométrie et triangle rectangle
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- Cours sur les suites
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- Equation et mise en équation (Exos 4eme)
- Mise en équation (Exos 3eme)
- Probabilités
- Fonction Logarithme
- Equations différentielles
- Exercices calcul de pourcentages
- Formules dérivées
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