Those Fascinating Numbers 201
20 683
the fourth number which can be written as the sum of two cubes in two distinct
ways: 20 683 =
103 +273
=
193 +243
(see the number 1 729); moreover, it is the
second number satisfying this property if one specifies that the representations
a3
+
b3
=
c3
+
d3
are such that (a, b) = (c, d) = 1.
20 737 (89 · 233)
the smallest composite number n such that 2n−2 1 (mod n); the other
composite numbers n 109 satisfying this property are 93 527, 228 727, 373 457,
540 857, 2 231 327, 11 232 137, 15 088 847, 15 235 703, 24 601 943, 43 092 527,
49 891 487, 66 171 767, 71 429 177, 137 134 727, 207 426 737, 209 402 327,
269 165 561, 302 357 057, 383 696 711 and 513 013 327; there are however other
such numbers, namely 73 · 48544121 and 524287 · 13264529 (R.K. Guy [101],
A12);
the second number of the form
122k
+ 1 (here with k = 2); it is conjectured
that all numbers of the form
12n
+1, with n 2, are composite; since it is clear
that
12n
+ 1 can be factored if n has an odd number, it follows that any prime
number of the form
12n
+ 1, with n 2, must be of the form
122k
+ 1; the
following is a table of the factorizations of the numbers
122k
+ 1 for 1 k 8:
1221
+ 1 = 5 · 29
1222
+ 1 = 20737 = 89 · 233
1223
+ 1 = 429981697 = 17 · 97 · 260753
1224
+ 1 = 184884258895036417 = 153953 · 1200913648289
1225
+ 1 = 34182189187166852111368841966125057
= 769 · 44450180997616192602560262634753
1226
+ 1 = 36097 · 81281 · 69619841 · 73389730593973249
·77941952137713139794518937770197249
1227
+ 1 = 257 · P136
1228
+ 1 = 8253953 · 295278642689 · C258.
20 771
the second prime number p such that
5p−1
1 (mod
p2):
the only prime
numbers p
232
satisfying this congruence are 2, 20 771, 40 487, 53 471 161
and 1 645 333 507 (see Ribenboim [169], p. 347).
20 979
the smallest number which can be written as the sum of the squares of three
prime numbers in 17 distinct ways: 20 979 = 112 + 972 + 1072 = 132 + 1012 +
1032 = 172+372+1392 = 172+892+1132 = 192+432+1372 = 192+672+1272 =
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