40 Jean-Marie De Koninck

relations 1p+23 = 32, 25+72 = 34, 73+132 = 29, 27+173 = 712, 35+114 = 1222,

177 + 76 2713 = 21 063 9282, 1 4143 + 2 213 4592 = 657, 9 2623 + 15 312 2832 =

1137, 438+96 2223 = 30 042 9072 and 338+1 549 0342 = 15 6133 (see C. Levesque

[124]).

123

• the eighth number n such that n · 2n − 1 is prime (see the number 115).

124

• the only number besides 188 which cannot be written as the sum of less than

five distinct squares (R.K. Guy [101], C20);

• the second pseudoprime in base 5: the ten smallest pseudoprimes in base 5 are

4, 124, 217, 561, 781, 1 541, 1 729, 1 891, 2 821 and 4 123.

125

• the smallest Canada perfect number50, that is a number for which the sum of

the squares of its digits is equal to the sum of its proper divisors 1; thus

12 +22 +52 = 5+25; the only numbers having this property are 125, 581, 8 549

and 16 999 (see J.M. De Koninck & A. Mercier [60]).

126 (=2 · 32 · 7)

• the smallest (and perhaps the only one) S-perfect number with three distinct

prime factors: a number n is said to be S-perfect (or a Granville number)

if

∑

d|n, dn, d∈S

d = n, where S is the set of integers defined by 1 ∈ S and

2 ≤ n ∈ S if and only if

∑

d|n, dn, d∈S

d ≤ n (see J.M. De Koninck & A. Ivi´c

[52]).

127

• the smallest prime number that is preceded by 13 consecutive composite num-

bers; indeed, there are no prime numbers between 113 and 127;

• the fourth Mersenne prime: 127 =

27

− 1;

• the exponent of the

12th

Mersenne prime

2127

− 1 (Lucas, 1876);

• the only known number n for which the corresponding number (47n − 1)/46 is

prime (see Ribenboim [169], p. 353).

50Created on the occasion of the 125th anniversary of the Canadian Confederation.