Euclid Number

The $n$th Euclid number is defined by

$$E_n\equiv 1+\prod_{i=1}^n p_i,$$

where $p_i$ is the $i$th Prime. The first few $E_n$ are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, ... (Sloane's A006862). The largest factor of $E_n$ are 3, 7, 31, 211, 2311, 509, 277, 27953, ... (Sloane's A002585). The $n$ of the first few Prime Euclid numbers $E_n$ are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, ... (Sloane's A014545) up to a search limit of 700. It is not known if there are an Infinite number of Prime Euclid numbers (Guy 1994, Ribenboim 1996).

See also Smarandache Sequences

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996.

Sloane, N. J. A. Sequences A014545, A006862/M2698, and A002585/M2697, 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.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 35-37, 1991.