Suppose f(x) = x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 is a monic polynomial with integer coefficients. Let x_1, ..., x_n be all its (complex) roots. If it ha

Suppose f(x) = x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 is a monic polynomial with integer coefficients. Let x_1, ..., x_n be all its (complex) roots. If it happens that the absolute value of x_1 is larger than 1, but absolute values of all x_2, x_3, ..., x_n are smaller than 1, the powers of x_1 will be close to integers. The golden ratio (1+sqrt(5)/2 is a root of x^2 - x - 1, and the other root (1-sqrt(5))/2 is smaller than 1 in absolute value, so the observation follows. One way to prove the assertion is following: note that for large n, x_1^n + x_2^n + ... + x_3^n is very close in value to x_1^n, because all the other summands are very small in absolute value, since they start out smaller than 1, and they decrease exponentially to 0. But it just so happens that x_1^n + x_2^n + ... + x_3^n is an integer -- indeed, this is a symmetric polynomial with integer coefficients in x_1, ..., x_n, and every such polynomial has a unique expression as a polynomial in elementary symmetric polynomial with integer coefficients -- this is the fundamental theorem of symmetric polynomials[1]. But, if you plug the roots of a given polynomial into elementary symmetric polynomials, you'll just get the coefficients of the original polynomial times (-1)^k, that is, the original a_0, a_1, ..., a_{n-1} -- for example, a_0 is clearly the product (-1)^n x_1 * x_2 * ... * x_n, a_{n-1} is the sum (-1)^1 (x_1 + ... + x_n), a_{n-2} is a sum of two-products (-1)^2 (x_1 x_2 + x_1 x_3 + ... + x_{n-1} x_n) and so on. For concrete example, note that x_1^2 + ... + x_n^2 = (x_1 + ... + x_n)^2 - 2(x_1 x_2 + ... + x_{n-1} x_n) = a_{n-1} - 2 a_{n-2}, which is an integer. Try expressing x_1^3 + ... + x_n^3 i n terms of a_{n-1}, a_{n-2} and a_{n-3}. [1] - https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial#The_fundamental_theorem_of_symmetric_polynomials https://en.wikipedia.org/wiki/Elementary_symmetric_polynomia...