000RM.dvi

622 Strings of composites

23.1 Strings of consecutive composite numbers ......

It is well known that there are strings of consecutive composite numbers
of arbitrary lengths. For example, thennumbers

(n+1)!+2, (n+1)!+3,···,(n+1)!+(n+1)

are all composites. These numbers are, however, very large.
In the table below, we give the first string ofnconsecutive composite
numbers.

n first string ofncomposite numbers

38 ··· 10

524 ··· 28

790 ··· 96

13 114··· 126

17 524··· 540

19 888··· 906

21 1130··· 1150

33 1328··· 1360

35 9552··· 9586

43 15684··· 15726

51 19610··· 19660

71 31398··· 31468

85 155922··· 156006

95 360654··· 360748

111 370262··· 370372

113 492114··· 492226

117 1349534··· 1349650

131 1357202··· 1357332

147 2010734··· 2010880

153 4652354··· 4652506

179 17051708··· 17051886

209 20831324··· 20831532

The first string of 100 consecutive composite numbers begins with

370262.^1 These are significantly less than 101!.

(^1) It actually extends to 370372, with 111 composite numbers. The string beginning with 396734 just
misses by 1; it gives 99 consecutive composites.