Those Fascinating Numbers 303

20 291 270

• the sixth number n such that Eσ(n) := σ(n + 1) − σ(n) satisfies Eσ(n + 1) =

Eσ(n): here the common value of Eσ is 365 040, since σ(20291270) = 37374480,

σ(20291271) = 37739520 and σ(20291272) = 38104560 (see the number 693).

20 427 264 (= 29 · 32 · 11 · 13 · 31)

• the fourth solution of

σ(n)

n

=

7

2

(see the number 4 320).

20 511 392

• the smallest number which can be written as the sum of two fifth powers and

as the sum of four fifth powers: 20 511 392 = 35 + 295 = 45 + 105 + 205 + 285.

20 831 323

• the smallest prime number which is followed by at least 200 consecutive compos-

ite numbers (in fact here by exactly 209 composite numbers); see the number

370 261.

20 840 574

• the third solution of σ(n) = σ(n + 15) (see the number 26).

20 916 224

• the smallest number n such that 12! divides 1 + 2 + . . . + n (see the number

224).

20 996 011

• the exponent of the

40th

Mersenne prime

220 996 011

− 1 (a 6 320 430 digit num-

ber) discovered by Michael Shafer on November 17, 2003 using the programme

developed by G. Woltman (see the number 1 398 269).

21 063 928

• the third number whose square can be written as the sum of a cube and a

seventh power (see B. Poonen, E. Schaefer & M. Stoll [165]): here 21 063 9282 =

762713 + 177; it is indeed possible to prove that the diophantine equation x2 =

y3 + z7 has only three solutions191 in positive integers x, y, z with (x, y, z) = 1;

here is the table of these solutions:

191In

1995, Henri Darmon and Andrew Granville [42] proved that if p, q, r are three positive integers

such that

1

p

+

1

q

+

1

r

1, then equation

Axp

+

Byq

=

Czr,

where A, B, C are non zero integers,

has only a finite number of solutions in positive integers x, y, z with (x, y, z) = 1.