2 Jean-Marie De Koninck

4

• the smallest number r which has the property that each number can be written

in the form x1 2 + x2 2 + . . . + xr 2, where the xi’s are non negative integers; the

problem consisting in determining if, for a given integer k ≥ 2, there exists a

number r (depending only on k) such that equation

(∗) n = x1

k

+ x2

k

+ . . . +

xrk

has solutions for each number n, is due to the English mathematician E. Waring

who, in 1770, stated without proof that “each number is the sum of 4 squares,

of 9 cubes, of 19 fourth powers, and so on”; if we denote by g(k) the smallest

number r such that equation (∗) has solutions for each number n, Lagrange

proved in 1770 that g(2) = 4, Wieferich and Kempner proved around 1910

that g(3) = 9, while R. Balasubramanian, J.M. Deshouillers & F. Dress [12]

proved in 1986 that g(4) = 19; it is conjectured that g(k) =

2k

+

[(3/2)k]

− 2

(where [x] stands for the largest integer ≤ x) for each integer k ≥ 2; see L.E.

Dickson

[65])1;

hence by using this formula, we find that the values of g(k),

for k = 1, 2, . . . , 20, are respectively 1, 4, 9, 19, 37, 73, 143, 279, 548, 1079,

2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899 (see

the book of Eric Weisstein [201], p. 1917).

5

• the smallest Wilson prime: a prime number p is called a Wilson prime if it

satisfies the congruence (p − 1)! ≡ −1 (mod

p2):

the only known Wilson

primes are 5, 13 and 563; K. Dilcher & C. Pomerance [68] have shown that

there are no other Wilson primes up to 5 ·

108.

6

• the smallest perfect number: a number n is said to be perfect if it is equal to

the sum of its proper divisors, that is if σ(n) = 2n; the sequence of perfect

numbers starts as follows: 6, 28, 496, 8 128, 33 550 336, . . . ; a number n is

said to be k-perfect if σ(n) = kn: if we let nk stand for the smallest k-perfect

number, then n2 = 6, n3 = 120, n4 = 30 240, n5 = 14 182 439 040 and n6 =

154 345 556 085 770 649 600;

• the smallest unitary perfect number: a number n is said to be a unitary perfect

number if

d|n

(d,n/d)=1

d = 2n, where (d, n/d) stands for the greatest common divisor

of d and n/d; only five unitary perfect numbers are known, namely 6, 60, 90,

87 360 and 146 361 946 186 458 562 560 000 = 218 · 3 · 54 · 7 · 11 · 13 · 19 · 37 · 79 ·

109 · 157 · 313: this last number was discovered by C.R. Wall [198] (see also

R.K. Guy [101], B3);

1In

1936, S. Pillai [161] proved that if one writes

3k

=

q2k

+ r with 0 r

2k,

then g(k) =

2k + [(3/2)k] − 2 provided r + q ≤ 2k.