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.
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