Index of my earlier mathematical notes

Having switched to WordPress for all my mathematical notes from now on, my first post here is a list of links to some mathematical notes I made on social media in the past, now stored as pdf files on my website.

1. Oscillatory price behaviour in mathematical economics: a simple numerical example

2. How Big Does a Near-Light-Speed Impactor Have to be to Wipe Out Life on Earth?

3. A little finding regarding Mersenne prime numbers

4. A little finding regarding the middle-half Cantor set

5. Some pretty pictures of quadratic forms

6. Some interesting literature references on the behaviour of power series on their circle of convergence

7. Elementary but fun problems

8. Two ways of proving a famous result regarding Fermat primes, side by side

9. Examples of ‘Unlimited Register Machine’ programs for Turing-computable functions

10. How the ‘Riemann sphere’ makes the distinction between lines and circles disappear

11. A catalogue of proofs that various sets, functions and relations are ‘primitive recursive’

12. A formal mathematical language for constructing ‘machine-checkable’ proofs, with MANY examples…

13. Memo on continued fractions

14. Collection of rules, techniques and theorems for solving polynomial congruences

15. Mathematical Induction, Greatest Common Divisor, Euclidean Algorithm, and Diophantine Equations

16. Prime numbers

17. A quicker proof of the equivalence of the principle of induction and the well-ordering principle

18. M823 Revision problems for Chapter 1: The fundamental theorem of arithmetic

19. An example in which using zero as an actual divisor gives a unique and meaningful answer

20. Advanced Number Theory Note #1: The abelian group structure of arithmetical functions under Dirichlet convolution

21. M823 Revision problems for Chapter 2: Arithmetical functions and Dirichlet multiplication

22. The big oh notation and asymptotic equality of functions

23. Advanced Number Theory Note #2: Derivation and applications of some important asymptotic formulas

24. Using Eisenstein’s criterion to prove that a cyclotomic polynomial of prime order is always irreducible

25. Solution to a problem making use of properties of the ‘natural homomorphism’ from a group G onto a quotient group G/H

26. Interesting problem involving groups of order p^2 where p is prime

27. Advanced Number Theory Note #3: Partial sums of Dirichlet convolutions, with some applications

28. M823 Revision problems for Chapter 3: Averages of arithmetical functions

29. Advanced Number Theory Note #4: Reformulations of the prime number theorem using Chebyshev’s functions

30. Advanced Number Theory Note #5: Proof of an important ‘Tauberian theorem’ and some applications

31. The little oh notation

32. Advanced Number Theory Note #6: Reformulation of the prime number theorem using the Möbius function

33. M823 Revision problems for Chapter 4: Some elementary theorems on the distribution of prime numbers

34. The standard solution to Russell’s paradox in modern set theory, and why there can be no ‘universal set’

35. Advanced Number Theory Note #7: Non-vanishing of the Dirichlet L(1, χ) function for real non-principal χ

36. Advanced Number Theory Note #8: Dirichlet’s theorem on primes in arithmetic progressions

37. Cute proof that if r is a quadratic residue of an odd prime p, then so is p – r if p ≡ 1 (mod 4), but not if p ≡ 3 (mod 4)

38. M823 Revision problems for Chapter 5: Congruences

39. M823 Revision problems for Chapter 6: Finite abelian groups and their characters

40. M823 Revision problems for Chapter 7: Dirichlet’s Theorem on Primes in Arithmetical Progressions

41. M823 Revision problems for Chapter 9: Quadratic Residues and the Quadratic Reciprocity Law

42. Conversation with Tom Bailey about how a photon can have momentum even though it has zero mass

43. Using the ABC conjecture to prove Fermat’s Last Theorem

44. AQA MPC1 Notes

45. Ideal class group for the ring of integers in an imaginary quadratic field, Part I

46. A rule of thumb to indicate when the majority of reported findings in a field of study are bullshit

47. Celebrating Heisenberg’s birthday: a ‘blow by blow’ mathematical derivation of his famous uncertainty principle

48. Quick proof of the cyclic property of the cross products of unit basis vectors in spherical polar coordinates

49. Simple proof that a photon seems to slow down and stop as it approaches the event horizon of a spherically symmetric Black Hole

50. AQA MPC2 Notes

51. Using a Residue Theorem from Complex Analysis to solve an improper integral

52. Time dilation formula from metric of special relativity

53. Conversation with Marc Fleury about the incompatibility of quantum theory and general relativity

54. AQA MSB1 Notes

55. Simple proof that every nth root of unity is a primitive dth root of unity for some divisor d of n

56. AQA MPC3 Notes

57. AQA MPC4 Notes