Small Numbers  (2)

The number two 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 . A quantity taken to the power 2 is said to be squared. The number of times  a given binary number  is divisible by 2 is given by the position of the first , counting from the right. For example,  is divisible by 2 twice, and  is divisible by 2 zero times.

The only known solutions to the congruence

are summarized in the following table (OEIS A050259). M. Alekseyev explored all solutions below  on Jan. 27 2007, finding no other solutions in this range.

	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  satisfying

for , 3, ... are , 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."

REFERENCES:

Crump, J. "2n mod n = c." http://www.immortaltheory.com/NumberTheory/2nmodn.htm.

Crump, J. "2^n = 3 (mod n): New Solution: 10991007971508067." 22 Jan 2007. http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0701&L=nmbrthry&T=0&P=3310.

Daiev, V. "Problem 636: Greatest Divisors of Even Integers." Math. Mag. 40, 164-165, 1967.

Guy, R. K. "Residues of Powers of Two." §F10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 250, 1994.

Montgomery, P.-L. "New solution to 2^n == 3 (mod n)." 24 Jun 1999. http://listserv.nodak.edu/scripts/wa.exe?A2=ind9906&L=NMBRTHRY&F=&S=&P=1753.

Sloane, N. J. A. Sequences A036236 and A050259 in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 41-44, 1986.