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A sequence of positive integers is called to be complete if every sufficiently large positive integer
can be written as a sum of distinct terms of
, that is if the set
![]() | (1) |
contains all sufficiently large integers.
Let denote the number of elements
with
.
We have several general sufficient conditions for completeness:
Theorem ( [1] ). Let be a sequence of distinct integers. Let
. Suppose that
![]() | (2) |
and that
![]() | (3) |
for every real number ,
. Then
is complete.
Corolarry ( [2] ). Let ,
, be a polynomial with integral coefficients such that for every integer
there is an infinity of primes
with
. Then the sequence
, where
runs through the primes, is complete.
Theorem ( [3] ). Let be a sequence of positive integers satisfying
(1) for all
where
,
(2) for every , each residue class modulo
contains at least one integer which is a sum of distinct
’s.
Then is complete.
Corolarry ( [4] ). Let be a sequence of positive integers for which
(1) for each there exists an
such that
,
(2) for every integer there are only finitely many terms in
divisible bt
.
Let ,
, denote the sequence of those terms from
whose subscripts are in
. Then
is complete for every
.
Theorem ( [5] ). Let be an infinite sequence of positive integers satisfying
(1) for
,
(2) at least one element of every arithmetic progression is expressible as a sum of distinct terms of .
Then is complete.
Birch [6] answered in affirmative way a problem (cf. the Corolarry) posed by P.Erdös proving
Theorem. Let be coprime positive integers, and
where
and
are arbitrary and
is sufficiently large. Then
is complete.
Corollary. If are coprime positive integers, then the sequence
is complete.
[1] | Cassels, J. W. (1960). On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math., 21, 111-124. |
[2] | Porubský, Š. (1979). Sums of prime powers. Monatshefte für Mathematik, 86, 301-303. |
[3] | Folkman, J. (1966). On the representation of integers as sums of distinct terms from a fixed sequence. Can. J. Math., 18, 643-655. |
[4] | Porubský, Š. (1978). Sums of distinct terms from a fixed sequence. Nordisk. Mat. Tidskr., 25-26, 185-187. |
[5] | Erdös, P. (1962). On the representation of large integers as sums of distinct summands taken from a fixed set. Acta Arith., 7, 345-354. |
[6] | Birch, B. J. (1959). Note on a problem of Erdös. Proc. Camb. Philos. Soc., 55, 370-373. |
Cite this web-page as:
Štefan Porubský: Complete sequence.