The document discusses various mathematical curiosities and problems, including circle division by lines producing patterns of numbers, Euler's formula relating vertices, edges and faces of polyhedra, the increasing difficulty of brute-force searching as the number of bits increases, the unsolved Collatz conjecture about a sequence related to dividing odd numbers by 3 and even numbers by 2, and Tupper's self-referential formula producing a pattern when graphed.

1. Mathematical curiosities
David Barina
October 12, 2018
David Barina Mathematical curiosities October 12, 2018 1 / 11

2. Circle division by lines
1, 2, 4, 8, 16, . . .
David Barina Mathematical curiosities October 12, 2018 2 / 11

3. Circle division by lines
1, 2, 4, 8, 16, 31, 57, 99, 163, 256, . . .
David Barina Mathematical curiosities October 12, 2018 3 / 11

4. Circle division by lines
(Euler)
V − E + F = 2
f (n) =
n
4
+
n
2
+ 1
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5. Exhaustive search (brute-forcing)
16-bit computers (IBM PC-XT 286)
32-bit computers (i386)
64-bit computers (AMD64, Opteron)
128-bit registers (SSE, . . . )
2048-, 4096-bit RSA keys
David Barina Mathematical curiosities October 12, 2018 5 / 11

6. Exhaustive search (brute-forcing)
16-bit computers (IBM PC-XT 286)
32-bit computers (i386)
64-bit computers (AMD64, Opteron)
128-bit registers (SSE, . . . )
2048-, 4096-bit RSA keys
iterate through all values
16 bits :
David Barina Mathematical curiosities October 12, 2018 5 / 11

7. Exhaustive search (brute-forcing)
16-bit computers (IBM PC-XT 286)
32-bit computers (i386)
64-bit computers (AMD64, Opteron)
128-bit registers (SSE, . . . )
2048-, 4096-bit RSA keys
iterate through all values
16 bits : 0 secs
32 bits :
David Barina Mathematical curiosities October 12, 2018 5 / 11

8. Exhaustive search (brute-forcing)
16-bit computers (IBM PC-XT 286)
32-bit computers (i386)
64-bit computers (AMD64, Opteron)
128-bit registers (SSE, . . . )
2048-, 4096-bit RSA keys
iterate through all values
16 bits : 0 secs
32 bits : 16 secs
64 bits :
David Barina Mathematical curiosities October 12, 2018 5 / 11

9. Exhaustive search (brute-forcing)
16-bit computers (IBM PC-XT 286)
32-bit computers (i386)
64-bit computers (AMD64, Opteron)
128-bit registers (SSE, . . . )
2048-, 4096-bit RSA keys
iterate through all values
16 bits : 0 secs
32 bits : 16 secs
64 bits : cca 2180 years
David Barina Mathematical curiosities October 12, 2018 5 / 11

10. Exhaustive search (brute-forcing)
16-bit computers (IBM PC-XT 286)
32-bit computers (i386)
64-bit computers (AMD64, Opteron)
128-bit registers (SSE, . . . )
2048-, 4096-bit RSA keys
iterate through all values
16 bits : 0 secs
32 bits : 16 secs
64 bits : cca 2180 years
187 bits : energy = annual output of the Sun
David Barina Mathematical curiosities October 12, 2018 5 / 11

11. Exhaustive search (brute-forcing)
16-bit computers (IBM PC-XT 286)
32-bit computers (i386)
64-bit computers (AMD64, Opteron)
128-bit registers (SSE, . . . )
2048-, 4096-bit RSA keys
iterate through all values
16 bits : 0 secs
32 bits : 16 secs
64 bits : cca 2180 years
187 bits : energy = annual output of the Sun
256 bits : 5.9 × 1020 Suns
David Barina Mathematical curiosities October 12, 2018 5 / 11

12. Collatz problem
C(n) =
3n + 1, if n odd,
n/2, if n even.
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13. Collatz problem
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
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14. Collatz problem
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15. Collatz problem
0
50
100
150
200
250
300
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
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16. Collatz problem
N =
2A
3B
−
B
n=1
2αn
3n
11 = 210
/34
− 26
/34
− 23
/33
− 21
/32
− 20
/31
In what ways does the prime factorization of a affect the prime
factorization of a + 1?
up to ca. 266 (distributed search)
David Barina Mathematical curiosities October 12, 2018 10 / 11

17. Tupper’s self-referential formula
1
2
< mod
y
17
2−17 x −mod( y ,17)
, 2
k is the 543-digit integer 960 939 379 . . . 748 404 719
if one graphs the set of points (x, y) in 0 ≤ x < 106 and k ≤ y < k + 17
satisfying the inequality
David Barina Mathematical curiosities October 12, 2018 11 / 11

18. Tupper’s self-referential formula
1
2
< mod
y
17
2−17 x −mod( y ,17)
, 2
k is the 543-digit integer 960 939 379 . . . 748 404 719
if one graphs the set of points (x, y) in 0 ≤ x < 106 and k ≤ y < k + 17
satisfying the inequality
David Barina Mathematical curiosities October 12, 2018 11 / 11