## Complete Sequence

A Sequence of numbers is complete if every Positive Integer is the sum of some subsequence of , i.e., there exist or 1 such that

(Honsberger 1985, pp. 123-126). The Fibonacci Numbers are complete. In fact, dropping one number still leaves a complete sequence, although dropping two numbers does not (Honsberger 1985, pp. 123 and 126). The Sequence of Primes with the element prepended,

is complete, even if any number of Primes each are dropped, as long as the dropped terms do not include two consecutive Primes (Honsberger 1985, pp. 127-128). This is a consequence of Bertrand's Postulate.

See also Bertrand's Postulate, Brown's Criterion, Fibonacci Dual Theorem, Greedy Algorithm, Weakly Complete Sequence, Zeckendorf's Theorem

References

Brown, J. L. Jr. Unique Representations of Integers as Sums of Distinct Lucas Numbers.'' Fib. Quart. 7, 243-252, 1969.

Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. A Primer for Fibonacci Numbers. XII.'' Fib. Quart. 11, 317-331, 1973.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.