##
**
On Lower Order Extremal Integral Sets Avoiding Prime Pairwise Sums
**

###
Ram Krishna Pandey

Department of Mathematics

Indian Institute of Technology, Patna

Patliputra Colony, Patna - 800013

India

**Abstract:**

Let *A* be a subset of {1,2, ..., *n*} such that the sum of no
two distinct elements of *A* is a prime number. Such a subset is
called a prime-sumset-free subset of {1,2, ..., *n*}. A
prime-sumset-free subset is called an extremal prime-sumset-free
subset of {1,2, ..., *n*} if *A* ∪ {*a*} is not a
prime-sumset-free subset for any *a* ∈ {1,2, ..., *n*} \
*A*. We prove that if *n* ≥ 10 then there is no any extremal
prime-sumset-free subset of {1,2, ..., *n*} of order 2 and if
*n* ≥ 13 then there is no any extremal prime-sumset-free subset
of {1,2, ..., *n*} of order 3. Moreover, we prove that
for each integer *k* ≥ 2, there exists a *n*_{k}
such that if *n* ≥
*n*_{k}
then there does not exist any extremal prime-sumset-free
subset of {1,2, ..., *n*} of length *k*. Furthermore, for some
small values of *n*, we give the
orders of all extremal prime-sumset-free subset
of {1,2, ..., *n*}, along with an example of
each order and we give all extremal prime-sumset-free subsets of {1,2,
..., *n*} of orders 2 and 3 for *n* ≤ 13.

**
Full version: pdf,
dvi,
ps,
latex
**

Received March 16 2012;
revised version received June 1 2012.
Published in *Journal of Integer Sequences*, June 12 2012.

Return to
**Journal of Integer Sequences home page**