48 Jean-Marie De Koninck

157

• the largest prime number p 10 000 such that

(pp

+ 1)/(p + 1) is a prime

number; the only known prime numbers satisfying this property are 3, 5, 17

and 157;

• the second prime number equally distant, by a distance of 6, from the preceding

and following prime numbers: p36 = 151, p37 = 157 and p38 = 163 (the smallest

prime number satisfying this property is 53);

• the smallest odd number which is not the sum of four non zero distinct squares

(F. Halter-Koch; see R K. Guy [101], C20);

• the smallest number n such that φ(2n + 1) φ(2n): the sequence of numbers

satisfying this inequality begins as follows: 157, 262, 367, 412, 472, 487, 577,

682, 787, 877, 892, 907, 997, 1207, 1237, 1567, 1627, 1657,. . .

158

• the second solution of σ(n) = σ(n + 19) (see the number 34).

159

• the smallest number n such that the Moebius function µ takes successively,

starting with n, the values 1, 0, 1, 0: the sequence of numbers satisfying this

property begins as follows: 159, 247, 303, 339, 411, 413, 685, 721, 849, 949,

. . . (see the number 3 647).

161

• the second solution of σ(n) = σ(n + 30) (see the number 88).

162

• the only number n

1012

such that σ(n) = 2n + 39 (see the number 196).

163

• the largest Heegner number: a number d is called a Heegner number if d 0 is

square-free and if the factorization in the set Ed = {a + b

√

−d : 2a, 2b ∈ Z} is

unique up to a unit (in the cases d = 1 and d = 2, the set Ed is instead defined

by Ed = {a + b

√

−d : a, b ∈ Z}); there exist exactly nine Heegner numbers,

namely 1, 2, 3, 7, 11, 19, 43, 67 and 163 (see J.H. Conway & R.K. Guy [36]);

• the only solution “SIX” of the French cryptarithm

DIX2

−

SIX2

= SEIZE × 4,

2632

−

1632

= 10650 × 4

(see Michel Criton [39]);