from Wolfram MathWorld
(来源:网站编辑 2025-12-11 20:17)
文章正文
The number two (2) is the second positive integer and the first prime number. It is even,
and is the only even prime
(the primes other than 2 are called the odd
primes). The number 2 is also equal to its factorial
since
![2!=2]()
. A quantity taken to the power
2 is said to be squared. The number of times
![k]()
a given binary number
![b_n...b_2b_1b_0]()
is divisible by 2 is given by the position of
the first
![b_k=1]()
, counting from the right. For example,
![12=1100]()
is divisible by 2 twice, and
![13=1101]()
is divisible by 2 zero times.
The only known solutions to the congruence
![2^n=3 (mod n)]()
are summarized in the following table (OEIS A050259). M. Alekseyev explored all solutions below
![10^(16)]()
on Jan. 27 2007, finding no other solutions in
this range.
![n]()
reference 4700063497 Guy (1994) 3468371109448915 M. Alekseyev (pers. comm., Nov. 13, 2006) 8365386194032363 Crump (pers. comm., 2000) 10991007971508067 Crump (2007) 63130707451134435989380140059866138830623361447484274774099906755 Montgomery (1999)
In general, the least
![n]()
satisfying
![2^n=k (mod n)]()
for
![k=2]()
, 3, ... are
![n=3]()
, 4700063497, 6, 19147, 10669, 25, 9, 2228071, ... (OEIS
A036236).
The chorus of the song "Power of Two" by the Indigo Girls (Swamp Ophelia, 1994), includes the lines "Adding up the total of a love that's
true; Multiply life by the power of two."
