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On the minimal points of perfect septenary quadratic forms
Published online by Cambridge University Press: 26 February 2010
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The object of this paper is to prove the following:
Theorem. Every perfect septenary quadratic form assumes its minimum value at a set of 7 points with integer co-ordinates whose determinant is 1.
This is true also, as shown by Rankin [1], with n ≤ 6 in place of 7. The proof will be shortened considerably by using the weaker result obtained in [1] for n = 7, and we shall also use the following classical results, see, e.g., [2], for Hermite's constant γn:
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