Collatz

Sīve mathēmaticā dē quaestiōne nōn solūtā

Annō 1937 Lotharus Collatz, mathēmaticus theodiscus, prīmum quaesīvit, an quīlibet numerus nātūrālis reducī ad ūnum per algorismum sequentem possit:

• Sī numerus pār est, dīvidātur per duo;

• Sī dispār, triplicētur et ad eum addātur ūnum;

Exemplī grātiā numerus 5 dispār est, itaque triplicandus est et ad eum ūnum addendum, quō factō invenimus numerum 16, quī dīvidī potest usque ad ūnum: 16, 8, 4, 2, 1.

Paulus Erdős, mathēmaticus Hungarus, dīxit mathēmaticam fortasse nōndum praestō esse hīs problēmatibus.

Nēmō enim quaestiōnem solvit.

Ēnōdātiō numerī 27

«Testūdō: Quīn cōnāris ā numerō 27 incipere? Nihil, scīlicet, polliceor. Sed aliquandō cōnāre animī causā. Praetereā suāderem ut magnam plagulam chartae afferrēs»¹.

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

Saepenumerō hominēs algorismum comparāre solent cum grānis grandinis, quī iterum iterumque in tempestāte atrōcī cadunt et surgunt.

Quīn ūtimur repraesentātiōnibus conjectūrae Collatz ut piīs imāginibus?

Aliī numerōs ab algorismō generātōs cum arboribus comparant.

Exemplum algorismī linguā pȳthōnicā scrīptum

import sys

def collatz(num):
    while num != 1:
        if num % 2 == 0:
            num = num // 2
            print(num)
        else:
            num = 3 * num + 1
            print(num)
try:
    while True:
        try:
            input_num = int(input('Scrībe numerum:\n'))
            collatz(input_num)
        except ValueError:
            print('Numerus integer tibi scrībendus est!')
except KeyboardInterrupt:
    sys.exit()

1. D. R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid (1979), Basic Books, 1999, p. 401