Sylvester's Sequence

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The sequence defined by 👁 e_0=2
and the quadratic recurrence equation

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

(Vardi 1991). Amazingly, there is a constant

(OEIS A076393) such that

(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 👁 e_n
satisfy

In addition, if 👁 0<x<1
is an irrational number, then the 👁 n
th term of an infinite sum of unit fractions used to represent 👁 x
as computed using the greedy algorithm must be smaller than 👁 1/e_n
.

The 👁 n
of the first few prime 👁 e_n
are 0, 1, 2, 3, 5, ..., corresponding to 2, 3, 7, 43, 3263443, ... (OEIS A014546). Vardi (1991) gives a lists of factors less than 👁 5×10^7
of 👁 e_n
for 👁 n<=200
and shows that 👁 e_n
is composite for 👁 6<=n<=17
. Furthermore, all numbers less than 👁 2.5×10^(15)
in Sylvester's sequence are squarefree, and no squareful numbers in this sequence are known (Vardi 1991).

See also

Cahen's Constant, Euclid's Theorems, Greedy Algorithm, Quadratic Recurrence Equation, Squarefree, Squareful

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References

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 " to the Rescue." §5.1 and 5.2 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 82-89, 1991.

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Sylvester's Sequence

Cite this as:

Weisstein, Eric W. "Sylvester's Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SylvestersSequence.html

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