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I like Erdos but I have to wonder how much of what he did matters if you look at results in binary instead of Base 10. It seems a lot of number theory seems to

I like Erdos but I have to wonder how much of what he did matters if you look at results in binary instead of Base 10. It seems a lot of number theory seems to gloss over this basic fact. In number theory, it seems Wolfram's four categories explain "impossible problems" pretty simply.

ajkjk 11y ago
What are you talking about? What base you write results in has nothing to do with number theory.
monochromatic 11y ago
This is not a result that has anything to do with base representation, nor are the vast majority of number theoretic results.
- Natsu 11y ago
  We're being trolled, I think. The only thing I can think of where base (sort-of) matters is that we have a formula for the Nth hexadecimal digit of pi, but not for the Nth decimal digit of pi. But I think that's due to lack of interest in making such a function, rather than some inherent reason why this could not be done.
  - byron_fast 11y ago
    I'm not really trying to troll, though it sounds that way. See below, for an example from Erdos' biography.
GFK_of_xmaspast 11y ago
What does that even mean.
ubernostrum 11y ago
Number theory deals with numbers, not the representations of them. The properties of numbers dealt with in number theory exist independently of the base chosen in which to represent the numbers; in fact, a result from number theory (the basis representation theorem) is the formalization of what a "base" actually is and how to represent a number in an arbitrary base.
- kenbellows 11y ago
  Number theory is like the metaphysics of maths. It's metamaths.
  - chestervonwinch 11y ago
    No, metamath is. https://en.wikipedia.org/wiki/Metamathematics https://en.wikipedia.org/wiki/Metamathematics
  - threatofrain 11y ago
    That's set or category theory.
- byron_fast 11y ago
  Your reply is what I always understood about number theory, and is true for many cases. But reading a number of important findings in number theory, when you look at the problem in binary - as Wolfram shows - every problem looks the same. The unsolvable problems are always noise, the solved problems have a pattern. Like a class 3/4 automata, any unsolved math problem is simply an elaborate way of reaching the same conclusion: randomness, where you can't predict any given value but must compute it.
  - disconcision 11y ago
    can you provide an example of an interesting problem in number theory which is dramatically transformed by approaching it in binary representation?
dogecoinbase 11y ago
Would you be willing to elaborate? I'd be curious to hear a specific example of a result due to Erdős that no longer matters in binary, and also a specific impossible problem that's explained by Wolfram (with some detail so I can follow along, of course).
- byron_fast 11y ago
  Erdos did a lot of work with primes, which if you're searching for patterns looking in binary makes a lot more sense than base 10. For an example of base-10 centrism that I'm talking about (not specifically Erdos, but from the biography "The Man Who Loved Only Numbers" on page 106 there is a discussion of Stan Ulam writing integers in a spiral and finding primes connected by diagonals. What possible meaning does this pattern have outside of base 10? This is the part of number theory that confuses me: why should such pattern finding matter, outside of base 10?
  - jfarmer 11y ago
    The pattern in the Ulam spiral is independent of the base we choose. They're just labels at regular intervals along a particular path. The fact that 3 is located at that particular point in the spiral is because its the third natural number, not because we write it as "3" in base-10. It's the 3rd natural number whether we write it as "3" (base 10), "11" (base 2), "10" (base 3), "|||", "三", a pictogram consisting of a hand holding up three fingers, or anything else. In short, it's the _spiral_ that dictates the pattern, not how we label the points on the spiral. As for what the pattern "means", well, it's hard to say and it might just our mind trying to make order where there is none. On the other hand, there are number theoretical conjectures that would account for certain aspects of the pattern, showing that it's more than just a psychological phenomenon. Are you sure you're not trolling? I guess a troll wouldn't answer.
    - byron_fast 11y ago
      Thank you for answering. In the example given in the book, Ulam creates rows of 10, so the base is significant to how the diagonals are found. Yes I understand the basis of number theory is to abstract away such things as the base. But the number of times I've read about number theorists and how they found their way to some pattern or truth that seems to depend on base 10 seems weird. So it is with this Ulam example: why is this a valid way to investigate the problem? It only seems valid in base 10 to me.

ajkjk 11y ago

What are you talking about? What base you write results in has nothing to do with number theory.

monochromatic 11y ago

This is not a result that has anything to do with base representation, nor are the vast majority of number theoretic results.

Natsu 11y ago
We're being trolled, I think. The only thing I can think of where base (sort-of) matters is that we have a formula for the Nth hexadecimal digit of pi, but not for the Nth decimal digit of pi. But I think that's due to lack of interest in making such a function, rather than some inherent reason why this could not be done.
- byron_fast 11y ago
  I'm not really trying to troll, though it sounds that way. See below, for an example from Erdos' biography.

GFK_of_xmaspast 11y ago

What does that even mean.

ubernostrum 11y ago

Number theory deals with numbers, not the representations of them. The properties of numbers dealt with in number theory exist independently of the base chosen in which to represent the numbers; in fact, a result from number theory (the basis representation theorem) is the formalization of what a "base" actually is and how to represent a number in an arbitrary base.

kenbellows 11y ago
Number theory is like the metaphysics of maths. It's metamaths.
- chestervonwinch 11y ago
  No, metamath is. https://en.wikipedia.org/wiki/Metamathematics https://en.wikipedia.org/wiki/Metamathematics
- threatofrain 11y ago
  That's set or category theory.
byron_fast 11y ago
Your reply is what I always understood about number theory, and is true for many cases. But reading a number of important findings in number theory, when you look at the problem in binary - as Wolfram shows - every problem looks the same. The unsolvable problems are always noise, the solved problems have a pattern. Like a class 3/4 automata, any unsolved math problem is simply an elaborate way of reaching the same conclusion: randomness, where you can't predict any given value but must compute it.
- disconcision 11y ago
  can you provide an example of an interesting problem in number theory which is dramatically transformed by approaching it in binary representation?

dogecoinbase 11y ago

Would you be willing to elaborate? I'd be curious to hear a specific example of a result due to Erdős that no longer matters in binary, and also a specific impossible problem that's explained by Wolfram (with some detail so I can follow along, of course).

byron_fast 11y ago
Erdos did a lot of work with primes, which if you're searching for patterns looking in binary makes a lot more sense than base 10. For an example of base-10 centrism that I'm talking about (not specifically Erdos, but from the biography "The Man Who Loved Only Numbers" on page 106 there is a discussion of Stan Ulam writing integers in a spiral and finding primes connected by diagonals. What possible meaning does this pattern have outside of base 10? This is the part of number theory that confuses me: why should such pattern finding matter, outside of base 10?
- jfarmer 11y ago
  The pattern in the Ulam spiral is independent of the base we choose. They're just labels at regular intervals along a particular path. The fact that 3 is located at that particular point in the spiral is because its the third natural number, not because we write it as "3" in base-10. It's the 3rd natural number whether we write it as "3" (base 10), "11" (base 2), "10" (base 3), "|||", "三", a pictogram consisting of a hand holding up three fingers, or anything else. In short, it's the _spiral_ that dictates the pattern, not how we label the points on the spiral. As for what the pattern "means", well, it's hard to say and it might just our mind trying to make order where there is none. On the other hand, there are number theoretical conjectures that would account for certain aspects of the pattern, showing that it's more than just a psychological phenomenon. Are you sure you're not trolling? I guess a troll wouldn't answer.
  - byron_fast 11y ago
    Thank you for answering. In the example given in the book, Ulam creates rows of 10, so the base is significant to how the diagonals are found. Yes I understand the basis of number theory is to abstract away such things as the base. But the number of times I've read about number theorists and how they found their way to some pattern or truth that seems to depend on base 10 seems weird. So it is with this Ulam example: why is this a valid way to investigate the problem? It only seems valid in base 10 to me.