Those Fascinating Numbers 59

223

• the smallest prime factor of the Mersenne number

237

− 1 (discovered by Fer-

mat), whose complete factorization is given by

237

− 1 = 223 · 616 318 177;

• the smallest prime number q whose corresponding Mersenne number

2q

− 1 has

exactly six prime factors; if we denote by qk the smallest prime number q such

that

2q

− 1 has exactly k prime factors, then we have the following table:

k

2qk

− 1

1

22

− 1 = 3

2

211

− 1 = 23 · 89

3

229

− 1 = 233 · 1103 · 2089

4

2157

− 1 = 852133201 · 60726444167 · 1654058017289 · 2134387368610417

5

2113

− 1 = 3391 · 23279 · 65993 · 1868569 · 1066818132868207

6

2223

− 1 = (see the number 18287 for the complete factorization)

7

2491

− 1 = (see the number 983 for the complete factorization)

8

2431

− 1 = (see the number 863 for the complete factorization)

9

2397

− 1 = (see the number 2383 for the complete factorization)

224

• the largest number which is not the sum of five distinct squares (G. Pall [158]);

where, say, a and c are odd while b and d are even. Then,

a2

−

c2

=

d2

−

b2,

(a − c)(a + c) = (d − b)(d + b). (1)

Let k = (a − c, d − b), so that there exist two integers and m such that

a − c = k, d − b = km, (, m) = 1. (2)

Since a − c and d − b are even, then k is even. By substituting (2) in (1), and dividing both sides

by k, it follows that

(a + c) = m(d + b). (3)

Since and m are co-prime, a + c must be divisible by m, in which case there exists a number α

such that

a + c = mα. (4)

Substituting (4) in (3) yields

d + b = α. (5)

From relations (4) and (5), it follows that α is the largest common divisor of a + c and d + b, and

is therefore even. The factorization of n is thus given by

n =

k

2

2

+

α

2

2

(m2

+

2),

since this last expression is equal to

1

4

(k2

+

α2)(m2

+

2)

=

1

4

(

(km)2

+

(k)2

+

(αm)2

+

(α)2

)

=

1

4

(

(d −

b)2

+ ((a −

c)2

+ (a +

c)2

+ (d +

b)2

)

=

1

4

(2a2

+

2b2

+

2c2

+

2d2)

=

1

4

(2n + 2n) = n.