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).
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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.
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