Mathematics and logic

1. Infinity of primes
2. Arbitrarily long sequences of successive integers, all not primes
3. Number of primes between 1 and n
4. Euler’s formula yields primes for x=0,1,2,3,…39
5. Irrational numbers: Algebraic, Transcendental (transcends operations of ordinary arithmetic)
6. Irrationality of square root of 2
7. Covering intervals
8. Euler’s constant C:
9. Approximating irrationals by rational numbers
10. Cantor’s existence proof of transcendental numbers
11. Non-constructibility of cube root of 2
12. Impossibility of finding center of circle with straightedge alone
13. Impossibility of covering modified chessboard with dominoes
14. Impossibility of decomposing cube into smaller cubes all of different size
15. Sperner’s Lemma: enumeration of patterns, fixed-point theorem follows
16. 292 ways of changing a dollar
17. The number system
18. The number of ways of partitioning a number into sums
19. The number of ways of partitioning a number into squares
20. Coin tossing: probability of m heads in n tosses
21. DeMoivre - Laplace Theorem
22. Axioms of probability theory equivalent to axioms of measure theory
23. Independent events implies normal distribution
24. Permutation group and solution of algebraic equations
25. Group of residues modulo p, Wilson’s Theorem
26. Homology group (a factor group)
27. Vectors, matrices, and geometry
28. Special theory of relativity as an example of geometric view in physics
29. Transformations, flows, and ergodicity
30. Iteration and composition of transformations: Markov chains
31. Consider two real valued functions both defined and continuous on the surface of a sphere. There must exist at least one point such that at this point and its antipode, both functions assume the same value.
32. Continuous, nowhere differentiable function
33. Convolution integrals: Heaviside calculus
34. Groups: braids. Does an algorithm exist to decide if two braids are equivalent? Yes, but general word problem in group theory is unsolved.
35. Gödels’s Theorem, Gödel numbering
36. Turing machine
37. Proof of independence of 5th postulate in plane geometry
38. Existence of sets satisfying axioms of set theory (including axiom of choice) but in which the continuum is of a “very high” power. Then sets intermediate between aleph-null and power of the continuum exist.
39. Maxwell’s equations
40. Ehrenfest game
41. Queues
42. Game theory by von Neumann
43. Information theory