Those Fascinating Numbers 195
16 796
the tenth Catalan number (see the number 14).
16 843
the smallest Wolstenholme prime: a prime number p satisfying the congruence
2p 1
p 1
1 (mod
p4)
is called a Wolstenholme prime: the only other known
Wolstenholme prime is 2 124 679 (R.J. McIntosh
[134])156.
16 999 (= 89 · 191)
the largest Canada perfect number:
12
+
62
+
92
+
92
+
92
= 89 + 191 (see the
number 125).
17 163
the largest number which is not the sum of the squares of distinct prime num-
bers (R.E. Dressler, L. Pigno & R. Young [70]).
17 280
the smallest number 2 which is equal to the product of the factorials of its
digits in base 8: 17 280 = [4, 1, 6, 0, 0]8 = 4! · 1! · 6! · 0! · 0!; the only known
numbers satisfying this property are
1, 2, 17 280, 348 364 800,
7059169943884597995687903472911168886283104071293337600000000000000000000,
185303211026970697386807466163918183264931481871450112000000000000000000000,
1909935803645696421452561061966062423003190467414438094886536676937305197064
and 306556928000000000000000000000000000000000000;
the following table re-
veals, for each base b with 5 b 12, the smallest number n which is equal to
the product of the factorials of its digits in base b:
interesting to make the following observations. It follows from Wilson’s Theorem that (156It)is
2p−1
1 (mod p) for all primes p and all integers n 1. In 1819, Babbage observed that
(2pp−11)

p−1
1 (mod
p2)
for each prime number p 2. In 1862, Wolstenholme established that
(
2p−1
p−1
)
1 (mod p3) for each prime number p 3.
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