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Old 2014-11-04, 01:07   #5
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Oct 2004

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The larger you choose d, the larger the algebraic polynomial degree. Each relation requires choosing (a,b) coordinates to plug into that larger polynomial, and as d increases the resulting values increase in size more and more quickly as the (a,b) values move away from the origin. So you essentially have a limited region in the (a,b) plane where the average size of the polynomial values are small enough that smooth relations are found often. As the number to be factored increases in size, the size of that nice region also increases and can support a (very slightly) larger d.

Thus, there is an optimal value of d that balances the rapidly increasing size of polynomial values with the temporary improvement in the smoothness probability. Many online papers go through the mathematics needed to determine that optimal d.
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