Sylvester’s sequence

Construct an Egyptian fractionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath equal to 1.


The denominators form the sequence 2, 3, 7, 43, 1807, … This is Sylvester’s sequence (listed in A58 of Sloane’s On-Line Encyclopedia of Integer Sequences), after the mathematician James Joseph Sylvester. The sequence can be calculated from the recurrence relationMathworldPlanetmath an=1+(an-1)2-an-1, with a0=2. Knowing the terms up to n-1 one can calculate an with the formula


If the sequence was meant to construct an Egyptian fraction equal to 2, then it would be 1, 2, 3, 7, 43, 1807, … and could still be calculated by multiplying the previous terms and adding 1, but the recurrence relation given above would have to be reformulated.

Whatever the definition, the sequence consists of coprimeMathworldPlanetmath terms, and thus can be used in Euclid’s proof of the infinity of primes. For this reason, these numbers are sometimes called Euclid numbers.

This sequence is useful in finding solutions to Znám’s problem.

Title Sylvester’s sequence
Canonical name SylvestersSequence
Date of creation 2013-03-22 15:48:09
Last modified on 2013-03-22 15:48:09
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A55
Synonym Euclid numbers
Synonym Sylvester sequence