372 Jean-Marie De Koninck
29 844 570 422 669
the number of prime numbers
1015.
29 996 224 275 697
the 1 000 000 000
000th
prime number (see the number 541).
31 065 037 602 817
the smallest prime factor of the Fermat number F17
=2217
+ 1 (see the number
70 525 124 609).
34 525 900 789 931
the smallest number which can be written in two distinct ways as the sum of
two numbers each with index of composition larger than 6: indeed,
34 525 900 789 931 =
24
·
313
·
53
+
1113
=
230
·
53
·
132
+
314
·
195,
where λ(24 · 313 · 53) 6.43387, λ(1113) = 13, λ(230 · 53 · 132) 6.31791 and
λ(314 · 195) 7.44556 (see the number 371 549).
42 600 227 803 225
the smallest perfect square m6 2 for which there exist numbers m1, m2, m3, m4
and m5 such that mi 2 (mi 1)2 = mi−1 2 for i = 2, 3, 4, 5, 6: here
42 600 227 803 225 = 6 526
8852
= 6 526
8842
+
36132
= 6 526
8842
+
36122
+
852
= 6 526
8842
+
36122
+
842
+
132
= 6 526
8842
+
36122
+
842
+
122
+
52
= 6 526
8842
+
36122
+
842
+
122
+
42
+
32.
48 689 748 233 307
the 24th powerful number n such that n + 1 is also powerful: here
48689748233307 =
33
·
1392
·
96612
and 48689748233308 =
22
·
73
·
132
·
432
·
3372
(see the number 288).
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