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).
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