Those Fascinating Numbers 119

1 080

• the smallest solution of

σ(n)

n

=

10

3

: the only solutions n

109

of this equation

are 1 080, 6 048, 6 552, 435 708, 4 713 984 and 275 890 944.

1 085

• the only solution “MORE” of the first known cryptogramme (or cryptarithm)

SEND + MORE = MONEY,

9 567 + 1 085 = 10 652,

sent by the English ludologist Henri Ernest Dudiney (1857-1945) to its editor

(see Michel Criton [39]).

1 089

• the smallest number 9 which is a proper divisor of the number obtained

by reversing its digits: the numbers satisfying this property are 1 089, 10 989,

109 989, 1 099 989, . . . while their doubles are 2 178, 21 978, 219 978, 2 199 978,. . .

1 091

• the seventh prime number p such that

(3p

− 1)/2 is itself a prime number; the

prime numbers p 10 000 such that

(3p

− 1)/2 is also prime are 3, 7, 13, 71,

103, 541, 1 091, 1 367, 1 627, 4 177, 9 011 and 9 551.

1 093

• the

smallest121

Wieferich prime: a prime number p is called a Wieferich

prime122

if it satisfies the congruence

2p−1

≡ 1 (mod

p2);

the only other known Wieferich

prime is 3 511; R. Crandall, K. Dilcher & C.Pomerance [37] proved that there

are no other Wieferich primes 4·1012, while computations done by volunteers

through the Internet have established that there are no other Wieferich primes

1.25 · 1015.

121It

is of some interest to display the complete factorization of

21 092

− 1, a 329 digit number:

21 092

− 1 =

32

· 5 ·

72

·

132

· 29 · 43 · 53 · 79 · 113 · 127 · 157 · 313 · 337 · 547 · 911 ·

10932

·1249 · 1429 · 1613 · 2731 · 3121 · 4733 · 5419 · 8191 · 14449

·21841 · 121369 · 224771 · 503413 · 1210483 · 1948129 · 22366891

·108749551 · 112901153 · 23140471537 · 25829691707

·105310750819 · 467811806281 · 4093204977277417 · 8861085190774909

·556338525912325157· 275700717951546566946854497

·86977595801949844993· 292653113147157205779127526827

·3194753987813988499397428643895659569.

122In

1909, Wieferich proved that if the first case of Fermat’s Last Theorem is false for a certain

prime number p (that is if xp +yp = zp, where p does not divide xyz), then p satisfies the congruence

2p−1

≡ 1 (mod

p2).

However, Meissner was the first (in 1913) to observe that the prime number

p = 1 093 does indeed satisfy this property.