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单词 ExamplesOfRangesOfConsecutiveIntegersForErdHosWoodsNumbers
释义

examples of ranges of consecutive integers for Erdős-Woods numbers


The most famous example of 16 as an Erdős-Woods number (http://planetmath.org/ErdHosWoodsNumber) is the range of 16 consecutive integers starting with 2184.

Another n for k=16 is 2044224, which we obtained by multiplying 2184 by 936. The factorization is 2044224=26×33×7×132, while 2044224+16=2044240=24×5×11×23×101. The table of factorizations

shows that each of the numbers in this range shares at least one factor with one if not both of the numbers capping the range.

Next we have a slightly longer example, this one for k=34. The smallest matching n is 47563752566, a squarefreeMathworldPlanetmath number with a factorization of 2×11×17×23×41×157×859. The number capping the end of the range is the decidedly non-squarefree 47563752600, with a factorization of 23×32×52×7×13×17×19×29×31. While the size of these numbers forbids verification on your typical pocket calculator, these numbers are well within the reach of a Javascript implementation of trial divisionMathworldPlanetmath. Here we could be tempted to omit the even numbersMathworldPlanetmath, as they obviously share 2 as a prime factorMathworldPlanetmath with the range start and the range end, as well as multiplesMathworldPlanetmath of 3 or 5 as they thus share factors with the range end. But, on the hope that it turns out to be at least a little bit instructive, the factorizations of all the numbers in our chosen range is given.

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更新时间:2025/5/4 7:05:07