PlanetMath: Sylvester's sequence

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Sylvester's sequence

(Definition)

Construct an Egyptian fraction equal to 1.

$\displaystyle {1 \over 2} + {1 \over 3} + {1 \over 7} + {1 \over {43}} + {1 \over {1807}} + \cdots$

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 relation $a_n = 1 + (a_{n - 1})^2 - a_{n - 1}$ , with . Knowing the terms up to one can calculate with the formula

$\displaystyle a_n = 1 + \prod_{i = 0}^{n - 1} a_i$

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 coprime 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.

"Sylvester's sequence" is owned by Mravinci.

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Other names:

Euclid numbers, Sylvester sequence

Attachments:: derivation of recurrence for Sylvester's sequence (Derivation) by rspuzio
sum of reciprocals of Sylvester's sequence (Proof) by rspuzio
Cahen's constant (Definition) by Mravinci

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Cross-references: Znám's problem, solutions, numbers, primes, infinity, proof, coprime, calculate, terms, recurrence relation, On-Line Encyclopedia of Integer Sequences, sequence, denominators, egyptian fraction
There are 8 references to this entry.

This is version 3 of Sylvester's sequence, born on 2006-03-24, modified 2006-10-02.
Object id is 7765, canonical name is SylvestersSequence.
Accessed 1840 times total.

Classification:

AMS MSC:

11A55 (Number theory :: Elementary number theory :: Continued fractions)

Pending Errata and Addenda

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