Abstract
Let L1, L2, L3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound offor integral values of u, v, w, not all zero. I proved a few years ago (1) thatmore precisely, thatexcept when L1, L2, L3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t3+t2-2t-1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to(in any order), where λ1,λ2,λ3 are real number whose product is In this case, L1L2L3|λ1λ2λ3 is a non-zero integer, and the minimum of its absolute value is 1, giving
Publisher
Cambridge University Press (CUP)
Reference8 articles.
1. (2) Journal London Math. Soc. 17 (1942), 107–15.
2. (9)This may be proved as in (8); the cyclotomic values of θ′, ø′, ψ′ are θ′ = 2 cos π/9, ø′ = 2 cos 5π/9, ψ′ = 2 cos 7π/9.
3. (8)It is easily verified that − øψ − 1 or −θ−1 − 1 satisfies the same cubic equation as θ, and since it lies between − 2 and − 1, it must equal ψ. Alternatively, the result follows from the fact that θ = 2 cos 2π/7, ø = 2 cos 4π/7, ψ = 2 cos 6π/7.
4. (3) Journal London Math. Soc. 16 (1941), 98–101.
5. On a conjecture of Mordell concerning binary cubic forms
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