“Sylvester's Sequence”的概念、定义、习题解题方法及翻译-数学辞典

单词 Sylvester's Sequence

释义

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 constructinga 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 torepresent

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 Squarefulnumbers in this sequence are known (Vardi 1991).

See also Euclid's Theorems, Greedy Algorithm, Squarefree, Squareful

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. SequenceA000058/M0865in ``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.

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