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.
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