Consecutive Smooth Numbers

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The smoothness of a number is its maximal prime factor. The number has smoothness 83, has smoothness 37, and has smoothness 97. All three numbers can be considered to be 97-smooth as well, since and . Multiplied out, these numbers are 7496643, 7496644, and 7496645, making this a consecutive 97-smooth triplet.

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For a given prime , all consecutive -smooth pairs can be found via Størmer's method [1]. For this Demonstration, all 97-smooth pairs were found. The number 97 is the prime. There are subsets of these primes. The products of these subsets form the value in the Pell equation . All of these equations were analyzed, with 21805 prime subsets yielding solutions. For smoothnesses above 97, solutions will be missing.

The largest proven maximal consecutive smooth pair is and [2].

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Contributed by: Ed Pegg Jr (March 2016)
Open content licensed under CC BY-NC-SA


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References

[1] Wikipedia. "Størmer's Theorem." (Mar 10, 2016) en.wikipedia.org/wiki/St% C3 % B8rmer's_theorem.

[2] D. Eppstein, "Smooth Pairs," LiveJournal (blog). (Mar 23, 2007) 11011110.livejournal.com/97325.html?thread=351533.



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