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Euclid Number


Euclid's second theorem states that the number of primes is infinite. The proof of this can be accomplished using the numbers

E_n=1+product_(i=1)^(n)p_i
(1)
=1+p_n#,
(2)

known as Euclid numbers, where p_i is the ith prime and p_n# 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 n of the first few prime Euclid numbers E_n 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 E_(13494), and it is not known if there are an infinite number of prime Euclid numbers (Guy 1994, Ribenboim 1996).

The largest factors of E_n for n=1, 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

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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.

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Euclid Number

Cite this as:

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

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