A polynomial with integer coefficients maps integers to integers. A rational function (which is a quotient of two polynomials) tends to map integers to rational numbers. Other than the trivial case where the denominator is a factor of the numerator, are there any rational functions that map all integers to integers? What is known about them and how can they be found?

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Vasquez · Accepted Answer

Let r(z)=p(z)q(z) such that r:Z→Z and p,q polynomials. Then q(z)|p(z) for all zϵZ and deg⁡p≥deg⁡q(check what happens at ∞). So if q is monic, then you can divide the polynomials, so p(z)=f(z)q(z)+g(z) with f,g polynomials and deg⁡g&amp;lt;deg⁡q. Then if g≠0,r(z)=f(z)+g(z)q(z), so g(z)q(z) is also an &quot;integer&quot; rational function, which is absqrt. So g≡0 and r is a integer polynomial. If q is non-monic, you can apply the theorem of the linked answer: call q0 the leading coefficient of q, then q0kp(z)=f(z)q(z)+g(z). Consider the integer rational function q0kr(z)=q0kp(z)q(z)=f(z)+g(z)q(z). As above, this implies that g(z)≡0. So r(z)=p(z)q(z)=q0kp(z)q0kq(z)=f(z)q(z)q0kq(z)=f(z)q0k. Then again r is a integer polynomial.

A polynomial with integer coefficients maps integers to integers. A rational function (which is a qu

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