The exponent to which a given quantity is raised is known as its Power. The expression is therefore known as `` to the th Power.'' The rules for combining quantities containing powers are called the Exponent Laws.

Special names given to various powers are listed in the following table.

Power | Name |

Square Root | |

Cube Root | |

2 | Squared |

3 | Cubed |

The Sum of th Powers of the first Positive Integers
is given by Faulhaber's Formula,

where is the Kronecker Delta, is a Binomial Coefficient, and is a Bernoulli Number.

Let be the largest Integer that is not the Sum of distinct th powers of Positive Integers (Guy 1994). The first few values for , 3, ... are 128, 12758, 5134240, 67898771, ... (Sloane's A001661).

Catalan's Conjecture states that 8 and 9 ( and ) are the only consecutive Powers (excluding 0 and 1), i.e., the only solution to Catalan's Diophantine Problem. This Conjecture has not yet been proved or refuted, although R. Tijdeman has proved that there can be only a finite number of exceptions should the Conjecture not hold. It is also known that 8 and 9 are the only consecutive Cubic and Square Numbers (in either order). Hyyrö and Makowski proved that there do not exist three consecutive Powers (Ribenboim 1996).

Very few numbers of the form are Prime (where composite powers need not be considered, since ). The only Prime Numbers of the form for and Prime correspond to , i.e., , , , .... The only Prime Numbers of the form for and Prime correspond to with , 2, 4, 6, 10, 14, 16, 20, 24, 26, ... (Sloane's A005574).

There are no nontrivial solutions to the equation

for (Guy 1994, p. 153).

**References**

Barbeau, E. J. *Power Play: A Country Walk through the Magical World of Numbers.*
Washington, DC: Math. Assoc. Amer., 1997.

Beyer, W. H. ``Laws of Exponents.'' *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, p. 158, 1987.

Guy, R. K. ``Diophantine Equations.'' Ch. D in *Unsolved Problems in Number Theory, 2nd ed.*
New York: Springer-Verlag, pp. 137, 139-198, and 153-154, 1994.

Ribenboim, P. ``Catalan's Conjecture.'' *Amer. Math. Monthly* **103**, 529-538, 1996.

Sloane, N. J. A. Sequences
A001661/M5393
and A005574/M1010
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Spanier, J. and Oldham, K. B. ``The Integer Powers and '' and ``The Noninteger Powers .''
Ch. 11 and 13 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 83-90 and 99-106, 1987.

© 1996-9

1999-05-26