Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plxCapcha has a deprecated constructor in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.capcha.php on line 9

Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plxErreur has a deprecated constructor in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.erreur.php on line 9

Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plxGlob has a deprecated constructor in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.glob.php on line 10

Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plxRecord has a deprecated constructor in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.record.php on line 10

Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plxMotor has a deprecated constructor in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.motor.php on line 10

Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plxFeed has a deprecated constructor in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.feed.php on line 9

Warning: Declaration of plxFeed::prechauffage() should be compatible with plxMotor::prechauffage($mode = '', $motif = '', $bypage = '') in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.feed.php on line 500

Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plxShow has a deprecated constructor in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php on line 9

Deprecated: Non-static method plxDate::microtime() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.motor.php on line 57

Deprecated: Non-static method plxUtils::getGets() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.motor.php on line 58

Deprecated: Non-static method plxUtils::mobileDetect() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.motor.php on line 66

Warning: Cannot modify header information - headers already sent by (output started at /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.capcha.php:9) in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.utils.php on line 548

Warning: Cannot modify header information - headers already sent by (output started at /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.capcha.php:9) in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.utils.php on line 549

Deprecated: Non-static method plxDate::dateToIso() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.motor.php on line 468

Warning: Cannot modify header information - headers already sent by (output started at /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.capcha.php:9) in /homepages/3/d170528928/htdocs/jm/bn/index.php on line 51

Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plxStats has a deprecated constructor in /homepages/3/d170528928/htdocs/jm/bn/plugins/stats/stats.php on line 9

Deprecated: Non-static method plxUtils::getIp() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/plugins/stats/stats.php on line 130

Deprecated: Non-static method plxUtils::getIp() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/plugins/stats/stats.php on line 131

Strict Standards: date(): We selected 'Europe/London' for 'BST/1.0/DST' instead in /homepages/3/d170528928/htdocs/jm/bn/plugins/stats/stats.php on line 142

Deprecated: Non-static method plxUtils::write() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/plugins/stats/stats.php on line 98
Linenn - Jeux et Mathematiques - <br /> <b>Deprecated</b>: Non-static method plxUtils::strCheck() should not be called statically in <b>/homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php</b> on line <b>127</b><br /> Les nombres gelés d'Antoine de Saint-Exupéry - linenn


Deprecated: Non-static method plxUtils::strCheck() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php on line 381
Les nombres gelés d'Antoine de Saint-Exupéry

Par
Deprecated: Non-static method plxUtils::strCheck() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php on line 393
jp, le
Deprecated: Non-static method plxDate::dateIsoToHum() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php on line 405

Deprecated: Non-static method plxDate::getCalendar() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.date.php on line 109

Deprecated: Non-static method plxDate::getCalendar() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.date.php on line 110
mardi 22 décembre 2009 à
Deprecated: Non-static method plxDate::dateIsoToHum() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php on line 417

Deprecated: Non-static method plxDate::getCalendar() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.date.php on line 109

Deprecated: Non-static method plxDate::getCalendar() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.date.php on line 110
19:22 |
Deprecated: Non-static method plxUtils::strCheck() should not be called statically in /homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php on line 448
Arithmétique | Page principale Jeux et Mathématiques

Deux triangles rectangles pour un nombre congruent donné

Antoine de Saint-Exupéry disparut le 31 juillet 1944. Quelques jours avant, le 15 juillet, il posa le problème suivant à son ami le capitaine Max Gelée.

Un parallélipipède rectangle dont la hauteur est égale à la diagonale du rectangle de base est exactement constitué par des dés cubiques de 1 cm de côté. La surface du rectangle de base est égale au produit de 311850 par un nombre premier inconnu.
Calculer la hauteur du parallélipipède.

Le texte de ce problème semblait disparu, il a été retrouvé et résolu par L. G. Vidiani après une longue et minutieuse enquête.

parallélépipède

La solution est bien évidemment de trouver un triangle Pythagorique c.-à-d. un triangle rectangle dont les trois côtés a, b, c soient entiers et dont l'hypoténuse c soit égale 311850 p = 2 × 34×52×7 × 11.

On sait que l'ensemble des triangles rectangles à côtés entiers est engendré par deux paramètres entiers positifs r et t, (r ≥ t) :
a=2rt, b = r2 - t2 et c = r2 + t2
On transforme ab=311850 p pour obtenir rt(r2 - t2)= 34×52×7 × 11 × p

Si on suppose p impair alors r, t et r2 - t2 sont impairs, ce qui est absurde.
On en conclut que le seul nombre premier convenable pour p ne peut être que 2.

En parcourant à l'aide d'un programme la moitié des 180 diviseurs de 22 × 34×52×7 × 11, on obtient les deux seules solutions, à une permutation près de a, b, c :
1) a=756, b=825, c=1119 et
2) a=540, b=1155, c=1275.
Les nombres « gelés » (ou de Gelée) sont donc 1119 et 1275.

Ce texte en PDF

Les nombres congruents sur jeux et mathématiques

Pour chercher :
les nombres gelés en commençant à 311850 (précédemment résolu pour 311850) et pour lesquels on doit aussi multiplier par 2. Ainsi pour 311904, il y a un seul nombre gelé qui est 1140 alors que pour 314496 il y en a deux qui sont 2040 et 2920.
311850 1275 1119, 311904 1140, 312120 1326, 312564 1525, 313344 2320, 313500 1205, 314244 1345, 314364 4625, 314496 2040 2920, 314646 1145, 315000 1625, 315276 1865, 315900 2817, 316350 3425, 317130 1891, 317400 1150, 317520 1722, 317856 1145 3700, 318270 1339, 319176 1130, 319200 1649, 319410 1443 1131, 319740 1241, 320166 1155, 320760 1818, 321024 1480, 321300 1371, 322896 1550, 322920 1614, 322944 1160, 323136 3270, 323730 5941, 324480 1352, 324870 1603, 325080 1289, 325314 1445, 325734 1165, 326370 1189, 326430 1479, 328440 4762, 328536 1170, 328560 1258, 328950 1237, 330330 1199 2167, 330480 2313, 330750 1365, 331350 1175, 332010 1157, 332310 3029, 332640 2652, 332820 1763, 333270 1219, 333396 1575, 333450 1191, 333564 1235, 334176 1180, 334854 4215, 335160 2534, 336000 2260 1480 1160, 336336 2030, ...
les suivants sans les solutions
336960 337014 337080 337500 337920 338520 339150 339480 339660 339864 340560 340704 341250 341880 342240 342726 342930 343470 344064 345000 345600 346560 347490 347820 348480 348486 348840 349680 349860 349920 350784 351000 351384 351624 352350 353010 353400 353430 353736 354144 354294 354354 354816 354900 355446 355680 355740 356430 357000 ...

les nombres gelés dans l'ordre croissant (selon le même principe)
5 10 13 15 17 20 25 26 29 30 34 35 37 39 40 41 45 50 51 52 53 55 58 60 61 65 68 70 73 74 75 78 80 82 85 87 89 90 91 95 97 100 101 102 104 105 106 109 110 111 113 115 116 117 119 120 122 123 125 130 135 ...
puis de 1100 à 1200 (sans omission)
1100 1102 1105 1107 1108 1109 1110 1111 1113 1114 1115 1117 1118 1119 1120 1122 1124 1125 1129 1130 1131 1135 1138 1139 1140 1144 1145 1147 1148 1150 1153 1154 1155 1156 1157 1158 1159 1160 1164 1165 1166 1167 1168 1170 1172 1173 1175 1180 1181 1182 1183 1184 1185 1186 1189 1190 1191 1192 1193 1195 1196 1199 1200 ...


Ecrire un commentaire




Quelle est la dernière lettre du mot loua ? :