## Sylvester's Sequence

The sequence defined by and the Recurrence Relation

 (1)

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
 (2)

(Vardi 1991). Amazingly, there is a constant
 (3)

such that
 (4)

(Vardi 1991, Graham et al. 1994). The first few numbers in Sylvester's sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... (Sloane's 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, .... 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).

References

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. Sequence A000058/M0865 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.

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.