Euclid Number
Euclid's second theorem states that the number of primes is infinite. The proof of this can be accomplished using the numbers
known as Euclid numbers, where 
is the 
th prime and 
is the primorial.
The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, ... (OEIS A006862; Tietze 1965,
p. 19).
The indices 
of the first few prime
Euclid numbers 
are 1, 2, 3, 4, 5, 11, 75, 171, 172,
384, 457, 616, 643, ... (OEIS A014545), so
the first few Euclid primes (commonly known as primorial
primes) are 3, 7, 31, 211, 2311, 200560490131, ... (OEIS A018239).
The largest known Euclid number is 
, and it
is not known if there are an infinite number of prime
Euclid numbers (Guy 1994, Ribenboim 1996).
The largest factors of 
for 
, 2, ... are
3, 7, 31, 211, 2311, 509, 277, 27953, ... (OEIS A002585).
SEE ALSO: Euclid-Mullin Sequence,
Euclid's Theorems,
Fortunate
Prime,
Integer Sequence Primes,
Primorial,
Primorial Prime,
Smarandache Sequences
REFERENCES:
Guy, R. K. Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.
Guy, R. and Nowakowski, R. "Discovering Primes with Euclid." Delta (Waukesha) 5,
49-63, 1975.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 28,
2003.
Naur, T. "Mullin's Sequence of Primes Is Not Monotonic." Proc. Amer.
Math. Soc. 90, 43-44, 1984.
Ribenboim, P. The
New Book of Prime Number Records. New York: Springer-Verlag, 1996.
Sloane, N. J. A. Sequences A006862/M2698, A002585/M2697, A014545,
and A018239 in "The On-Line Encyclopedia
of Integer Sequences."
Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity
to Modern Times. New York: Graylock Press, 1965.
Wagon, S. Mathematica
in Action. New York: W. H. Freeman, pp. 35-37, 1991.
Wagstaff, S. S., Jr. "Computing Euclid's Primes." Bull. Inst. Combin.
Appl. 8, 23-32, 1993.
Referenced on Wolfram|Alpha:
Euclid Number
CITE THIS AS:
Weisstein, Eric W. "Euclid Number." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EuclidNumber.html
Contact the MathWorld Team
30-level 12-ary tree