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Thus, to study the order of rational approximation of algebraic numbers of degree ~ 3 one needs other methods. The earliest result in that direction was obtained by Liouville (see [1844a], [1844bD. 6. 3. Let a E A, dega = n ~ 1. If a:f; p/q, then Proof. When n = 1 the theorem follows from (1). Let n ~ 2. If la-p/ql ~ 1, the inequality in the theorem holds for any c E (0,1). Suppose that la-p/ql < 1, in which case Ip/ql < lal + 1. Let P(z) = anz n + ... + ao be the minimal polynomial of a. Then Ip(~)1 = Ip(a) - p(~)1 = li;q P1(Z)dZI $ = la - ~I max Ip'(a + (J(~ q 0$89 q < la - I a)) < la - ~I ~ klakl(lal q ~ + l)k-l < ~I nL(a)(lal + l)n-l .
This argument, often called the "pigeonhole principle," is that if n+ 1 objects are placed in n pigeonholes, then at least one of the pigeonholes must contain two or more objects. We shall give several examples of theorems whose proofs use this principle. 8. Let al, ... ,aN E JR, XI, ... ,XN E N, and X = maxXj . Then there exists a nontrivial N -tuple Xl, ... , XN E Z ("nontrivial" means not all zero) such that §4. Stronger Versions of Liouville's and Thue's Theorems 41 j= 1, ... ,N. 9. Let M,N,X E N, N> M, N N =L Li Ai ~ Llai,jl, i ai,jXj, ai,j E JR, j=1 j=1 = 1, ...
Moreover, there is a formula that allows one to find any solution. To find this formula it is enough to determine the so-called "fundamental solution" Xl, YI. Then the other solutions are given by the formula n E No, where we keep in mind that (±x, ±y) is a solution whenever (x, y) is. 5). 3. The Case n ~ 3. , they differ by only a constant factor. For n ~ 3 we do not have this. However, we see from (13) that if we could replace Liouville's inequality by the inequality la - ~I ~ c(a)h(q)q-n, h(q) ---+ 00 as q ---+ 00 , §3.