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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 imp

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.
    - jfarmer 11y ago
      Without sharing the specific example, I don't know what to say. The Wikipedia page on the Ulam spiral is consistent with every representation/exposition I've seen and doesn't depend on the base we choose for the labels. I don't even know what "rows of 10" means in the context of the Ulam spiral.
      - byron_fast 11y ago
        Yes I think the example of Ulam spiral in the Erdos book is not very good. The Wiki page is clear. I wish I could find the example of Erdos work where I felt the same way... but the pile of his work is too immense. Still, though, my karma hasn't taken enough of a beating: viewing a result set in binary and comparing it to automata output still seems a shortcut to understanding. That's an idea that is unpopular for sure.
        jfarmer 11y ago
        There are definitely properties of numbers that are more easily stated in one base vs. another, but I don't see how an automata would matter. Any automata that can process strings in one base can be converted into an automata that can process strings in any other base. An example of such a property is that a real number in the interval [0,1] is a member of the Cantor set if and only if it has no 1's in its base 3 representation (not a typo, yes, base 3).
byron_fast 11y ago
For an example of using the Wolfram automata-view to look at a problem, think of Fermat's Last Theorem. Can a specific cube (or any higher power, really) be expressed as the sum of two other cubes? The automata view shows us the answer because when you look at the list of cubes (or any higher power) as binary the resulting image meets the Wolfram test for randomness. Looking at these numbers in any other base has no obvious meaning, but in binary you can instantly see whether some set at least looks like a random automata. When I look at number theory or indeed any insoluble math problem it always seems to be some variation on people trying to write an equation that satisfies something from a random automata. Essentially all unsolved problems boil down to an argument about whether you believe automata show randomness or not.

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.
  - jfarmer 11y ago
    Without sharing the specific example, I don't know what to say. The Wikipedia page on the Ulam spiral is consistent with every representation/exposition I've seen and doesn't depend on the base we choose for the labels. I don't even know what "rows of 10" means in the context of the Ulam spiral.
    - byron_fast 11y ago
      Yes I think the example of Ulam spiral in the Erdos book is not very good. The Wiki page is clear. I wish I could find the example of Erdos work where I felt the same way... but the pile of his work is too immense. Still, though, my karma hasn't taken enough of a beating: viewing a result set in binary and comparing it to automata output still seems a shortcut to understanding. That's an idea that is unpopular for sure.
      - jfarmer 11y ago
        There are definitely properties of numbers that are more easily stated in one base vs. another, but I don't see how an automata would matter. Any automata that can process strings in one base can be converted into an automata that can process strings in any other base. An example of such a property is that a real number in the interval [0,1] is a member of the Cantor set if and only if it has no 1's in its base 3 representation (not a typo, yes, base 3).

byron_fast 11y ago

For an example of using the Wolfram automata-view to look at a problem, think of Fermat's Last Theorem. Can a specific cube (or any higher power, really) be expressed as the sum of two other cubes? The automata view shows us the answer because when you look at the list of cubes (or any higher power) as binary the resulting image meets the Wolfram test for randomness. Looking at these numbers in any other base has no obvious meaning, but in binary you can instantly see whether some set at least looks like a random automata. When I look at number theory or indeed any insoluble math problem it always seems to be some variation on people trying to write an equation that satisfies something from a random automata. Essentially all unsolved problems boil down to an argument about whether you believe automata show randomness or not.