# Consecutive Smooth Numbers

Consecutive Smooth Numbers

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.

3×7×11×17×23×83

2

2

4

37

5×13×29×41×97

97>83

97>37

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 =33554432 subsets of these primes. The products of these subsets form the value in the Pell equation -2k=1. All of these equations were analyzed, with 21805 prime subsets yielding solutions. For smoothnesses above 97, solutions will be missing.

n

n

th

25

25

2

k

2

x

2

y

The largest proven maximal consecutive smooth pair is 83×103×107×139× and 5×13×19×43×59×61×71×89×109× [2].

3

2

2

23

2

47

2

151

2

173

3

3

2

7

3

167