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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.

nreference
4700063497Guy (1994)
3468371109448915M. Alekseyev (pers. comm., Nov. 13, 2006)
8365386194032363Crump (pers. comm., 2000)
10991007971508067Crump (2007)
63130707451134435989380140059866138830623361447484274774099906755Montgomery (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."


See also

1, Binary, 3, Ruler Function, Squared, Two-Ears Theorem, Two-Graph, Two-Sheeted Hyperboloid, Zero

Explore with Wolfram|Alpha

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.

Cite this as:

Weisstein, Eric W. "2." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/2.html

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