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I don't know much about number theory, but I think this visualization probably provides significant intuition as to why number theory, which contains only theor

I don't know much about number theory, but I think this visualization probably provides significant intuition as to why number theory, which contains only theorems about integers, uses tons of math that is about a lot more than integers. Edit: Why would anyone down-vote this comment. Barbarians at the gates, I tell you! The only possible explanation is that the down-voters thinks that the relationship between number theory and other kinds of math is so obvious that it needs no further explication to those who are less informed. In which case, this is the pinnacle of arrogance. Or the down-voter believes that there is no such relationship between number theory and other kinds of math, in which case this is the pinnacle of ignorance. In any case, I personally found the visualization inspiring. I.e., it makes me want to learn some more.

thlt 14y ago
this chart basically includes a set of periodic curves, even though it looks nice but what does it tell you about prime numbers ? Intersected by only two curves 1 and itself ??? well, everyone knows this, no need to make a chart.
- nessus42 14y ago
  It tells you that prime numbers and periodic curves might be related more than you might have thought at first blush. I.e., if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. As I mentioned, I don't know much about number theory, but I do know that it uses a lot of math that is counterintuitive at first blush. Perhaps this visualization gives us a clue as to why that is the case.
  - dxbydt 14y ago
    >if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. Dude, no offense, you are really making stuff up. The periodicity has to to with the divisors of the composites. Every prime p has exactly 2 curves - the wave of period 1, and the wave of period p. There's nothing interesting or useful to take way from that observation. otoh, you look at a composite c - it has several divisors & each divisor d generates a curve of period d, and those curves intersect in interesting ways...though I don't see how you could mathematically analyze them to tell you anything about the primes nearby. They are mostly pretty patterns, not mathematically useful...here are 2 quite famous & useful diagrams on periodicity in primes if you are interested in that sort of thing - 1. the prime number cross - http://img841.imageshack.us/img841/1329/primenumbercross.gif http://img841.imageshack.us/img841/1329/primenumbercross.gif 2. hippocampal neurons & primes - http://www.hindawi.com/journals/amp/2011/519178/fig8/ http://www.hindawi.com/journals/amp/2011/519178/fig8/
    - nessus42 14y ago
      I take it, then, that you find nothing inspirational in the videos made by Vi Hart either. To each, their own. As to the diagrams you pointed me at, they mean nothing to me, and do not inspire me. If I knew more about number theory, perhaps they would. Some things are not about information, they are about inspiration. The visualization in the OP provides inspiration that number theory is connected to other fields of math. You can see structure in the way that the periodic curves intersect and the primes are the gaps. If you could understand that structure, then maybe you could understand the gaps. Then again maybe not. That doesn't mean that the question and the visualization doesn't cause you to think and wonder. Sure, this is old hat to mathematicians, and for all I know, this approach is a complete dead end. Sometimes dead ends are interesting too.
    - Retric 14y ago
      It does give you a good intuition as to why twin primes aka N, N+2 are so common.
      - thlt 14y ago
        how so ?
        Retric 14y ago
        The number between them have a high number of small factors including 2 and 3. 2 * 3 = 6: 5 and 7 are prime. 2 * 3 * 2 = 12: 11 and 13 are prime 2 * 3 * 3 = 18: 17 and 19 are prime 2 * 3 * 5 = 30: 11 and 13 are prime 2 * 3 * 7 = 42: 41 and 43 are prime. However, 2 * 3 * 101 = 606 but 605 is not prime. But, 2 * 3 * 5 * 5 = 150 and 149 and 151 are prime. 2 * 3 * 2 * 3 * 5 = 180 and 179 and 181 are prime.
        jjaredsimpson 14y ago
        I don't see where the insight is. twin primes must be of the form (6k-1, 6k+1). So of course there will be a 2,3 at least. Smaller numbers have multiples that are more densely distributed among the integers.

thlt 14y ago

this chart basically includes a set of periodic curves, even though it looks nice but what does it tell you about prime numbers ? Intersected by only two curves 1 and itself ??? well, everyone knows this, no need to make a chart.

nessus42 14y ago
It tells you that prime numbers and periodic curves might be related more than you might have thought at first blush. I.e., if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. As I mentioned, I don't know much about number theory, but I do know that it uses a lot of math that is counterintuitive at first blush. Perhaps this visualization gives us a clue as to why that is the case.
- dxbydt 14y ago
  >if you can mathematically describe and analyze this set of periodic curves, then you have also described and can analyze the prime numbers. Dude, no offense, you are really making stuff up. The periodicity has to to with the divisors of the composites. Every prime p has exactly 2 curves - the wave of period 1, and the wave of period p. There's nothing interesting or useful to take way from that observation. otoh, you look at a composite c - it has several divisors & each divisor d generates a curve of period d, and those curves intersect in interesting ways...though I don't see how you could mathematically analyze them to tell you anything about the primes nearby. They are mostly pretty patterns, not mathematically useful...here are 2 quite famous & useful diagrams on periodicity in primes if you are interested in that sort of thing - 1. the prime number cross - http://img841.imageshack.us/img841/1329/primenumbercross.gif http://img841.imageshack.us/img841/1329/primenumbercross.gif 2. hippocampal neurons & primes - http://www.hindawi.com/journals/amp/2011/519178/fig8/ http://www.hindawi.com/journals/amp/2011/519178/fig8/
  - nessus42 14y ago
    I take it, then, that you find nothing inspirational in the videos made by Vi Hart either. To each, their own. As to the diagrams you pointed me at, they mean nothing to me, and do not inspire me. If I knew more about number theory, perhaps they would. Some things are not about information, they are about inspiration. The visualization in the OP provides inspiration that number theory is connected to other fields of math. You can see structure in the way that the periodic curves intersect and the primes are the gaps. If you could understand that structure, then maybe you could understand the gaps. Then again maybe not. That doesn't mean that the question and the visualization doesn't cause you to think and wonder. Sure, this is old hat to mathematicians, and for all I know, this approach is a complete dead end. Sometimes dead ends are interesting too.
  - Retric 14y ago
    It does give you a good intuition as to why twin primes aka N, N+2 are so common.
    - thlt 14y ago
      how so ?
      - Retric 14y ago
        The number between them have a high number of small factors including 2 and 3. 2 * 3 = 6: 5 and 7 are prime. 2 * 3 * 2 = 12: 11 and 13 are prime 2 * 3 * 3 = 18: 17 and 19 are prime 2 * 3 * 5 = 30: 11 and 13 are prime 2 * 3 * 7 = 42: 41 and 43 are prime. However, 2 * 3 * 101 = 606 but 605 is not prime. But, 2 * 3 * 5 * 5 = 150 and 149 and 151 are prime. 2 * 3 * 2 * 3 * 5 = 180 and 179 and 181 are prime.
        jjaredsimpson 14y ago
        I don't see where the insight is. twin primes must be of the form (6k-1, 6k+1). So of course there will be a 2,3 at least. Smaller numbers have multiples that are more densely distributed among the integers.