3 is the only integer which is the sum of the preceding positive integers (1+2=3) and the only number which is the sum of the factorials of the preceding positive integers (1!+2!=3). It is also the first odd prime. A quantity taken to the power 3 is said to be cubed.

The sequence 1, 31, 331, 3331, 33331, ... (OEIS A033175) consisting of n=0, 1, ... 3s followed by a 1 has its nth term is given by


The result is prime for n=1, 2, 3, 4, 5, 6, 7, 17, 39, ... (OEIS A055520); i.e., for 31, 331, 3331, 33331, 333331, 3333331, 33333331, ... (OEIS A051200), a fact which Gardner (1997) calls "a remarkable pattern that is entirely accidental and leads nowhere."

See also

1, 2, Collatz Problem, Cubed, Period Three Theorem, Ternary, Three-Arc Illusion, Three-Choice Polygon, Three-Choice Walk, Three-Colorable Graph, Three-Colorable Knot, Three Conics Theorem, Three Jug Problem, Three-Valued Logic, Trefoil Knot, Wigner 3j-Symbol, Zero

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Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer-Verlag, p. 194, 1997.Sloane, N. J. A. Sequences A033175, A051200, and A055520 in "The On-Line Encyclopedia of Integer Sequences."Smarandache, F. Properties of Numbers. University of Craiova, 1973.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 46-48, 1986.

Cite this as:

Weisstein, Eric W. "3." From MathWorld--A Wolfram Web Resource.

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