Those Fascinating Numbers 105

756

• the third number n for which gcd

2n

n

, 105 = 1; the sequence of numbers

satisfying this property begins as follows: 1, 10, 756, 757, 3160, 3187, 3250,

7560, 7561, 7651, 20007, . . . ; Ron Graham offers 100$ U.S. to anyone who

proves that this sequence is infinite (see R.K. Guy [101], B33);

• the tenth number n such that f(n) f(m) for all numbers m n, where

f(n) :=

Σ∗

1

p

, with the star indicating that the sum runs over all prime numbers

p n which do not divide

(

2n

n

)

(see the number 364); it is also the smallest

number n such that f(n) 1: here f(756) = 1.07698 . . . .

767

• the largest solution x of

y2

=

(x)

0

+

(x)

1

+

(x)

2

+

(x)

3

, namely (x, y) = (767, 8672)

(R.K. Guy [101], D3).

769

• the smallest prime factor of

1232

+ 1, whose complete factorization is given by

1232

+ 1 = 769 · 44450180997616192602560262634753;

it is conjectured that

12n

+ 1 is

composite109

for each n ≥ 2 (see the number

20 737).

770

• the number of digits in the decimal expansion of the 15th perfect number

21278(21279 − 1).

773

• the smallest odd number k such that k +

2n

is

composite110

for all n k: the

sequence of odd numbers k with this property begins as follows: 773, 2131,

2491, 4471, 5101, . . .

109It

is easy to establish that in order for

12n

+ 1 to be prime, one must have that n =

2m

for a

certain integer m ≥ 0, but that this condition is not suﬃcient.

110Here

in reality, 773 +

2n

is composite for all numbers n 955. On the other hand, one can

prove that 78 557 + 2n is composite for each n ≥ 1; indeed, this follows from the fact that at least

one of the primes 3, 5, 7 13, 19, 37, 73 divides

2n

+ 78 557 for all n ≥ 1 (see the number 78 557 for

a similar argument concerning the numbers of the form k · 2n + 1).