Question 974259
I don't think a polynomial like that exists, because
a {{{P(x)}}} such that {{{P(x + 3/2)=P(x)}}} is a periodic function (like trigonometric functions),
that has a period of {{{3/2}}} or {{{3/"2 n"}}} for a natural number {{{n}}} .
 
For such a function,
{{{P(23)=P(17)=687}}} ,
because {{{23-17=6=4*(3/2)}}} , so {{{P(23)=P17+4(3/2))}}} and
{{{P(17+4(3/2))=P(17+3(3/2))=P(17+2(3/2))=P(17+3/2)=P(17)=687}}}