Those Fascinating Numbers 99

646

• the number of Carmichael numbers

109

(Swift, 1975); if nk stands for the

number of

Carmichael103

numbers

10k,

then n4 = 7, n5 = 16, n6 = 43,

n7 = 105, n8 = 255, n9 = 646, n10 = 1 547, n11 = 3 605, n12 = 8 241,

n13 = 19 279, n14 = 44 706, n15 = 105 212 and n16 = 246 683.

649

• the smallest solution x of the Fermat-Pell equation

x2

−

13y2

= 1 (with y = 0):

this solution (x, y) is (649, 180).

650

• the fourth solution of σ(n) = 2n + 2 (see the number 464).

651

• the smallest number 1 whose sum of divisors is a tenth power: σ(651) = 210.

656

• the seventh dihedral perfect number (see the number 130).

657 (=

32

· 73)

• the ninth unitary hyperperfect number (see the number 288).

661

• the largest known number n such that An := n! − (n − 1)! + (n − 2)! − . . . +

(−1)n+11!

is prime: using a computer, one can check that An is prime for n = 3,

4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160 and 661 (see R.K. Guy [101], B43).

666

• the third number which is equal to the sum of its digits added to the sum of

the cubes of its digits: the only numbers satisfying this property are 12, 30,

666, 870, 960 and 1 998.

103This is in accordance with the result of W.R. Alford, A. Granville & C. Pomerance [3] according

to which not only the set of Carmichael numbers is infinite, but if C(x) stands for the number of

Carmichael numbers ≤ x, then C(x) x2/7 for x suﬃciently large; indeed, C(1011) = n11 = 3605

1389.5 =

(1011)2/7.

More recently, G. Harman [108] proved that, for x large enough, C(x)

xβ

,

where β 0.33. It is conjectured that, for any ε 0, C(x) x1−ε if x ≥ x0(ε).