A Strange Grid Reveals Hidden Connections Between Simple Numbers
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A Strange Grid Reveals Hidden Connections Between Simple Numbers
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Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#2Maybe I don't understand the full implications of the problem, but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table. Both multiplication and addition yield points on lines described by the usual mx+b equation. Addition constrains x to 1, so all of the results will be clustered near the origin, with more coincidences as an unavoidable result. Conversely, if x is allowed to exceed 1, the resulting 'lines' will spread out on the graph and leave more room for unique points.
What am I missing?
Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#3Maybe I don't understand the full implications of the problem, but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table. Both multiplication and addition yield points on lines described by the usual mx+b equation. Addition constrains x to 1, so all of the results will be clustered near the origin, with more coincidences as an unavoidable result. Conversely, if…
dunno, but it's neat to think about. multiplication can cover more ground in integer space than addition, so it can reach more unique places than addition for a given set of numbers. it's kinda like comparing how many squares knights and pawns can reach on a chessboard for a given number of moves.
Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#4Overview of the sum/product problem by Tao https://terrytao.wordpress.com/2007/12/06/milliman-lecture-i...
[e: that lecture is almost a decade older than the result mentioned in OP]
Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#5It's almost like one dimension of entropy has been exhausted.
Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#6Maybe I don't understand the full implications of the problem, but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table. Both multiplication and addition yield points on lines described by the usual mx+b equation. Addition constrains x to 1, so all of the results will be clustered near the origin, with more coincidences as an unavoidable result. Conversely, if…
Did you read the article? The numbers doesn't have to be consecutive, for example what you said is only true for arithmetic progresions, but for geometric progressions it happens the opposite that multiplications has less diversity. Read the full article.
Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#7Maybe I don't understand the full implications of the problem, but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table. Both multiplication and addition yield points on lines described by the usual mx+b equation. Addition constrains x to 1, so all of the results will be clustered near the origin, with more coincidences as an unavoidable result. Conversely, if…
”but it seems obvious that a multiplicative table will result in more 'numerical diversity' than a summation table”
That isn’t true. If you pick the set of numbers {1, 2, 4, 8}, you get 10 different sums (2, 3, 4, 5, 6, 8, 9, 10, 12, and 16) but only 7 different products (1, 2, 4, 8, 16, 32, and 64)
Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#8i dont get it...what is the hidden connection exactly?
Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#9i dont get it...what is the hidden connection exactly?
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Re: A Strange Grid Reveals Hidden Connections Between Simple Numbers
#10This kind of feels like it's touching on some of the same themes as the ABC conjecture.