Sorceress

Alternative Logarithms, Part 3

Mar 21st, 2020
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  1. Alternative Logarithms, Part 3
  2. ------------------------------
  3.  
  4. For the next bit of analysis, let us think about how we were constructing s() in part 2 -- We were defining s(primes), then deducing s(composites) using the original identity.
  5.  
  6. We observed that s() was well-defined: Because :
  7. (1) the fundamental theorem of arithmetic (x having unique factorisation);
  8. (2) f() was an associative and commutative operation (XOR in that example).
  9.  
  10. But addition is also an associative and commutative operation, as are all of the Lp-norms, so could s() be constructed in a similar way for these?
  11.  
  12. Indeed. Consider the addition operation: s(a.b) = s(a) + s(b), as seen in the original logarithm.
  13.  
  14. s(x) can be defined for prime x, and then deduced for composite x by adding the corresponding s(primes).
  15.  
  16. As in the XOR example, it doesn't seem to matter what value we assign to s(prime x), although to avoid degeneracy these values must not form a degenerate set (by which I mean linearly independent over the addition operation, with integer coefficients). ie, we must not have some combination of s(primes) equal to 0. One way to achieve this is with s(i'th prime) = T^i, where T is a transcendental.
  17.  
  18. x s(x)
  19. 1 0
  20. 2 1
  21. 3 T
  22. 4 s(2)+s(2) = 2
  23. 5 T^2
  24. 6 s(2)+s(3) = 1 + T
  25. 7 T^3
  26. 8 s(2)+s(2)+s(2) = 3
  27. 9 s(3)+s(3)= 2T
  28. 10 1 + T^2
  29. 11 T^4
  30. 12 2 + T
  31. ...
  32.  
  33. Furthermore, s(rational x) can be evaluated from observing that
  34.  
  35. s(a.1/a) = s(a) + s(1/a) = s(1) = 0 ==> s(1/a) = - s(a)
  36.  
  37. x s(x)
  38. 1/2 -1
  39. 2/3 1 - T
  40. 6/5 1 + T - T^2
  41. 8/12 3 - (2 + T) = 1 - T // s(8/12) = 1 - T = s(2/3)
  42. ...
  43.  
  44. Clearly every polynomial in the ring Z[T] is the image of some rational x. This is quite an interesting result, imo.
  45.  
  46. Some comments from part 2 still apply:
  47.  
  48. - The primes could appear in any order, not necessarily in the ascending order I have used.
  49.  
  50. - Each prime could correspond with a polynomial in T, rather than a single monomial.
  51.  
  52. - Specific degeneracies are entirely optional in this construction. eg, s() could easily be made to map some primes to zero, such that their presence in x is entirely ignored. And if we want to map some composite to zero, then s() of one of it's prime factors could be defined as a linear combination of s() of it's other prime factors such that that happens. Alternatively, T could be the root of a specific polynomial, such that the composite corresponding to that polynomial is mapped to zero.
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