  For earlier considered our sequence A166944 in [4] we prove three statements
of its connection with twin primes. We also give a sufficient condition for the
infinity of twin primes and pose several new conjectures; among them we propose
a very simple conjectural algorithm of constructing a pair $(p,\enskip p+2)$ of
twin primes over arbitrary given integer $m\geq4$ such that $p+2\geq m.$
