On Linear Diophantine Equations and Fibonacci Numbers

AbstractBy Minkowski′s theorem on linear forms it is shown that the homogeneous linear diophantine equation a · x = a1x1 + · · · + aKxK = 0 (K≥2, ai≠0, 1≤i≤K) has a non-zero integral solution x with r(x) ≤ (K−1) · r(a)1/(K−1) (where r(y) = ΠKi=1 max(|yi|, 1)). It turns out that it is very difficult to decide if the exponent 1/(K−1) is optimal or not. (Of course the case K=2 is trivial). It is shown that there exist integral coefficients a ∈ ZK with arbitrarily large r(a) such that every non-zero integral solution a · x = 0 satisfies r(x)≥cKr(a)1/K (log r(a))−K In the non-trivial case K = 3 coefficients like a = (F6m+1, F6m+2, F6m+2 + 1), where the Fn denote the usual Fibonacci numbers, can be used to prove the optimality of the exponent 1/(K−1). Furthermore the connection between this diophantine problem and the discrepancy of linear functions ƒ(t) = at mod 1 is discussed