Those Fascinating Numbers 183

9 871

• the largest prime number with four distinct digits; if qk stands for the largest

prime number made up of exactly k distinct digits, then q1 = 7, q2 = 97,

q3 = 983, q4 = 9 871, q5 = 98 731, q6 = 987 631, q7 = 9 876 413, q8 = 98 765 431

and q9 = 987 654 103; it is clear that q10 does not exist (see the number 1 039

for the similar question for the smallest prime number with k distinct digits).

9 901

• one of the only two prime numbers p (the other one is 101) satisfying the

property154 that any number of the form abcdefabcdef is divisible by p.

9 941

• the exponent of the

22nd

Mersenne prime

29 941

− 1 (Gillies, 1963).

9 973

• the largest four digit prime number.

10 007

• the smallest five digit prime number.

10 080

• the

21rst

highly composite number (see the number 180).

10 223

• the smallest candidate still “suspected” of being a Sierpinski number (see the

number 78 557).

10 243

• the smallest prime number made of five distinct digits (see the number 1 039).

10 305 (= 32 · 5 · 229)

• the smallest number n which allows the sum

m≤n

Ω(m)=4

1

m

to exceed 1; the smallest

number that allows this sum to exceed 2 is 1 799 938.

154To

obtain this result, see the footnote tied to the number 9091, observing that 1 000 001 =

101 · 9901.