Number theory fascinates me. I studied elementary number theory, the next step, analytical number theory requires knowledge of group and representatuon theory ( characters ) and complex analysis. I am working hard on that. In the meanwhile I like to study the famous but accessible problems of number theory. An interesting problem is the following.

It has been conjectured that the function $x_n = f(x_{n-1})$ with $x_0 \in \mathbf{N}$ always ends in $1,4,2,1,4,2,1 \cdots$ where

$$f: x \mapsto \begin{cases}

\frac{x}{2} \text{ if } x \equiv 0 \mod{2} \\

3x+1 \text{ if } x \equiv 1 \mod{2}

\end{cases}$$

Example:

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

A fornal proof that this sequence ends like this is unknown. According to Conway it is even undecidable. Many papers have been written on the problem and it has several dedicated websites.

Very similar, but nevertheless generally generating smaller sequences is the function

$$f: x \mapsto \begin{cases}

1 \text{ if } x = 2 \\

\text{ else } x \mapsto \begin{cases}

\frac{x}{4} \text{ if } x \equiv 0 \mod{4} \\

6x+2 \text{ if } x \equiv 1 \text{ or } 3 \mod{4} \\

\frac{x-2}{4} \text{ if } x \equiv 2 \mod{4}

\end{cases}

\end{cases}$$

27

164

41

248

62

15

92

23

140

35

212

53

320

80

20

5

32

8

2

1

1-2017 More on the randomness of randomness.

2 months ago

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