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The mathematical term for this is the probability of a number being b-smooth. Here ‘b’ is 2^32
by AnotherGoodName 4mo ago
The mathematical term for this is the probability of a number being b-smooth. Here ‘b’ is 2^32
- danbruc 4mo agoRelated but not strong enough. 17 x 17 x 17 = 4,913 is 2^8-smooth - no prime factors larger than 2^8 - and it is less than 2^16, but 17 x 17 = 289 does not fit into a byte. Smoothness is required but not sufficient for a product representation to exist.
- svat 4mo agoIt's related, but not the same thing. For example, for b=10, the number 70=2x5x7 is b-smooth, but it cannot be written as the product of two numbers less than b. Here are the other b-smooth (counter)examples for b=10: | n | factorization | products of two numbers |-----|----------------|------------------------------------ | 50 | 2 * 5^2 | 1x50, 2x25, 5x10 | 60 | 2^2 * 3 * 5 | 1x60, 2x30, 3x20, 4x15, 5x12, 6x10 | 70 | 2 * 5 * 7 | 1x70, 2x35, 5x14, 7x10 | 75 | 3 * 5^2 | 1x75, 3x25, 5x15 | 80 | 2^4 * 5 | 1x80, 2x40, 4x20, 5x16, 8x10 | 84 | 2^2 * 3 * 7 | 1x84, 2x42, 3x28, 4x21, 6x14, 7x12 | 90 | 2 * 3^2 * 5 | 1x90, 2x45, 3x30, 5x18, 6x15, 9x10 | 96 | 2^5 * 3 | 1x96, 2x48, 3x32, 4x24, 6x16, 8x12 | 98 | 2 * 7^2 | 1x98, 2x49, 7x14