Those Fascinating Numbers 11
(and no such number is knwon, yet!) a property which is clearly satisfied if n is
prime (see D. Borwein, J.M. Borwein, P.B. Borwein & R. Girgensohn [23]); the
smallest five Giuga numbers are
30 :
1
2
+
1
3
+
1
5

1
30
= 1,
858 :
1
2
+
1
3
+
1
11
+
1
13

1
858
= 1,
1 722 :
1
2
+
1
3
+
1
7
+
1
41

1
1722
= 1,
66 198 :
1
2
+
1
3
+
1
11
+
1
17
+
1
59

1
66198
= 1,
2 214 408 306 :
1
2
+
1
3
+
1
11
+
1
23
+
1
31
+
1
47057

1
2214408306
= 1;
the other six known Giuga numbers are
24 423 128 562 = 2 · 3 · 7 · 43 · 3041 · 4447,
432 749 205 173 838 = 2 · 3 · 7 · 59 · 163 · 1381 · 775807,
14 737 133 470 010 574 = 2 · 3 · 7 · 71 · 103 · 67213 · 713863,
550 843 391 309 130 318 = 2 · 3 · 7 · 71 · 103 · 61559 · 29133437,
244 197 000 982 499 715 087 866 346 = 2 · 3 · 11 · 23 · 31 · 47137 · 28282147
and 554 079 914 617 070 801 288 578 559 178
= 2 · 3 · 11 · 23 · 31 · 47059 · 2259696343 · 110725121051;
to this day only 11 Giuga numbers have been found; it is not known if there are
infinitely many; let us mention that surprisingly for each known Giuga number
n, we have

p|n
1
p

p|n
1
p
= 1;
the largest number n such that τ (n) = φ(n) (see the number 8);
the denominator of the Bernoulli number B4 =
1
30
; the sequence of Bernoulli
numbers (Bn)n≥0 is defined implicitly by the expansion
z
ez 1
=

n=0
Bnzn
n!
(see E.W. Weisstein [201], p.111).
31
the third Mersenne prime: 31 =
25
1 (Euler, 1750);
the sixth prime number pk such that p1p2 . . . pk + 1 is prime (see the number
379);
the smallest number n which allows the sum
i≤n
1
i
to exceed 4 (see the number
83);
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