202 Jean-Marie De Koninck

232 + 412 + 1372 = 292 + 372 + 1372 = 312 + 972 + 1032 = 372 + 592 + 1272 =

372 + 972 + 1012 = 472 + 832 + 1092 = 592 + 832 + 1032 = 612 + 672 + 1132 =

672 + 712 + 1072 = 712 + 732 + 1032 = 732 + 792 + 972.

21 147

• the ninth Bell number (see the number 52).

21 480

• the

22nd

number n such that n! − 1 is prime (see the number 166).

21 701

• the exponent of the 25th Mersenne prime 221 701 − 1 (Noll and Nickel, 1978).

21 978

• the fourth number (which is not a palindrome) which divides the number ob-

tained by reversing its digits (see the number 1 089).

22 020

• the smallest number n such that n, n+1, n+2, n+3, n+4, n+5 are all divisible

by a square 1: here 22 020 =

22·3·5·367,

22 021 =

192·61,

22 022 =

2·7·112·13,

22 023 =

32

· 2447, 22 024 =

23

· 2753, 22 025 =

52

· 881 (see the number 242).

22 073

• the second prime number q such that

∑

p≤q

p is a perfect square: here

∑

p≤22073

p = 25633969 =

50632:

the sequence of numbers satisfying this pro-

perty begins as follows: 23, 22 073, 67 187, 79 427, 10 729 219, . . .

22 434

• the largest solution x of the diophantine equation

x2

+ 19 =

yn:

J.H.E. Cohn

[35] proved that the only three solutions of this equation, with n ≥ 2 and x ≥ 0,

are (x, y, n) = (9, 10, 2), (18,7,3) and (22 434,55,5).

22 440

• one of the three numbers n such that the polynomial x5 −x±n can be factored;

the two others are n = 15 and n = 2 759 640: here x5 − x ± 22 440 = (x2 ∓

12x + 55)(x3 ± 12x2 + 89x ± 408) (see S. Rabinowitz [166]).