> How would one prove the below? > > If p(x) is a polynomial with integer coefficients, then the curve y=p(x) > cannot cross the x-axis at pi, pi + 1, pi^2, e, e+1 or e^2. > > Seems kind of unbelievable, because these curves hit all kinds of real > numbers, since they are continuous lines.
There are only countably many solutions to all of the polynomials in Z[x] (the algebraic numbers) while there are uncountable many real numbers. That is most of the numbers can't be algebraic or as it's said, the real numbers are almost everywhere transcendental (is not algebraic) or the real numbers are almost nowhere algebraic. On the other hand, unlike the integers, both the rationals and the algebraic are dense subsets of the reals. That's why you find infinitely many of them in every tiny interval.
However, proving a particular number, such as e or pi, is transcendental is a difficult matter.