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From: Olivier Miakinen <om+news@miakinen.net>
Newsgroups: fr.sci.maths
Subject: =?UTF-8?Q?Re:_Le_calcul_de_la_racine_carr=c3=a9..._pour_des_nuls_:?=
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Date: Sun, 10 Nov 2024 11:15:00 +0100
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Le 10/11/2024 10:43, efji a écrit :
> 
> * En plus de ceux déjà repéré : 999 = 10^3-1, 500500, 499500 = 
> 5*10^{2k+1} ± 5*10^k,
> on en trouve plein de rigolos :
> 4444444, 36363636, 38883889, 61116111, 63636364, 74747475, 88888888, 
> 432432432, 567567568, 3888938889, 4132841328, 4756047561, 6111061111, 
> 7979797980, 8888888889, 9090909091, 9132791328, 9756097560, 44444444445, 
> 55555555555, 324324324324, 425925425926, 590909590909, 593554593555, 
> 604247604247, 675675675676, 695156695156, 730769730769, 733414733415, 
> 740774077407, 749749749750, 750249750250, 804843804843, 831683168316, 
> 909090909090, 925925925925, 980518980519 avec presque l'impression que 
> la densité de rigolos augmente quand on monte :)

Je me suis amusé à regarder aussi les puissances troisième et quatrième en
plus des carrés, et (surtout si on s'autorise de compter les zéros initiaux)
on trouve des nombres rigolos qui fonctionnent dans les trois cas :

3888938889
 = √ 15123845682376554321
   = 1512384568+2376554321
 = ∛ 058815711625428924040757889369
   = 0588157116+2542892404+0757889369
 = ∜ 0228730708224339944007530377000663771041
   = 0228730708+2243399440+0753037700+063771041

4999950000
 = √ 24999500002500000000
   = 2499950000+2500000000
 = ∛ 124996250037499875000000000000
   = 1249962500+3749987500+0000000000
 = ∜ 0624975000374997500006250000000000000000
   = 0624975000+3749975000+0625000000+0000000000

5000050000
 = √ 25000500002500000000
   = 2500050000+2500000000
 = ∛ 125003750037500125000000000000
   = 1250037500+3750012500+0000000000
 = ∜ 0625025000375002500006250000000000000000
   = 0625025000+3750025000+0625000000+0000000000


D'autres exemples de nombres rigolos, en s'interdisant les zéros initiaux avant
découpage :

671671
 ∜ 203529043026143035282081 = 203529+043026+143035+282081

63636364
 √ 4049586823140496 = 40495868+23140496
 ∛ 257700981126972226596544 = 25770098+11269722+26596544

5867158672
 √ 34423550882424803584
 = 3442355088+2424803584
 ∜ 1184980855354889473501540380260979245056
 = 1184980855+3548894735+0154038026+0979245056

6666666667
 ∜ 1975308642370370370400000000000987654321
 = 1975308642+3703703704+0000000000+0987654321


-- 
Olivier Miakinen