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 =