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.