This sequence arises in Euclid's proof that there are an infinite number of primes. The proof proceeds by constructing
a sequence of primes using the recurrence
relation
(Aho and Sloane 1973, Vardi 1991, Graham et al. 1994). The first few numbers in Sylvester's sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... (OEIS
A000058). The satisfy
(5)
In addition, if
is an irrational number, then the th term of an infinite sum of unit fractions used to represent
as computed using the greedy algorithm must be
smaller than .
The
of the first few prime are 0, 1, 2, 3, 5, ..., corresponding to 2, 3, 7, 43, 3263443,
... (OEIS A014546). Vardi (1991) gives a lists
of factors less than of for and shows that is composite for . Furthermore, all numbers
less than
in Sylvester's sequence are squarefree, and no squareful numbers in this sequence are known (Vardi 1991).
Aho, A. V. and Sloane, N. J. A. "Some Doubly Exponential Sequences." Fib. Quart.11, 429-437, 1973.Finch,
S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, 2003.Graham,
R. L.; Knuth, D. E.; and Patashnik, O. Research problem 4.65 in Concrete
Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley,
1994.Sloane, N. J. A. Sequences A000058/M0865,
A014546, and A076393
in "The On-Line Encyclopedia of Integer Sequences."Finch,
S. R. "Quadratic Recurrence Constants." §6.10 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 443-448,
2003.Vardi, I. "Are All Euclid Numbers Squarefree?" and "PowerMod
to the Rescue." §5.1 and 5.2 in Computational
Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 82-89,
1991.
and the quadratic
recurrence equation
![E=1/2sqrt(6)exp{sum_(j=1)^infty2^(-j-1)ln[1+(2e_j-1)^(-2)]}=1.2640847353...](/images/equations/SylvestersSequence/NumberedEquation3.svg)
satisfy
is an irrational number, then the 
th term of an infinite sum of unit fractions used to represent
as computed using the greedy algorithm must be
smaller than 
.
of the first few prime 
are 0, 1, 2, 3, 5, ..., corresponding to 2, 3, 7, 43, 3263443,
... (OEIS A014546). Vardi (1991) gives a lists
of factors less than 
of 
for 
and shows that 
is composite for 
. Furthermore, all numbers
less than 
in Sylvester's sequence are squarefree, and no squareful numbers in this sequence are known (Vardi 1991).
