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.
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