A course of mathematical analysis by Khinchin A.

By Khinchin A.

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Example text

For k > 2 there is a partial affine plane of order η with k pencils of parallel lines if and only if there are k - 2 mutually orthogonal Latin squares of order n. Proof: The proof follows from the same construction as was done for Theorem 9. • Corollary. The finite partial affine planes of order η having four pencils of parallel lines correspond to the pairs of orthogonal Latin squares A and Β of order n. Thus the open question stated above can be rephrased in terms of affine geometry: Is there a partial affine plane of order 10 with 5 pencils of parallel lines?

Suppose that A and Β are orthogonal Latin squares of order η and that J and Κ are constant η Χ η matrices with entries j and k, respectively. Prove that A + J and Β + Κ are orthogonal Latin squares (again the addition should be done mod «). 5. Suppose that A and Β are orthogonal Latin squares of order n, and that k in Z„ is relatively prime to n. Show that A and kB are orthogonal Latin squares, where the (/th entry in kB is x „ b , . 6. Let π be an odd natural number. Show that A = [β„] and fi = [fc,--] are orthogonal Latin squares of order /j where ; and 2 / + / - 1.

In this language it was already seen that H(2) = H(6) = 1 and that H(n) ^ 2 otherwise. Results on affine planes in Chapter 4 will imply that whenever ρ is a prime number, H(p ) = p - 1. The connection between pencils of parallel lines and Latin squares is summarized in the following theorem: k k • THEOREM 9. For η :> 2, there is an affine plane of order η if and only if / / ( « ) > η - 1. Proof: Given an affine plane of order n, select two of its (n + 1) pencils of parallel lines, and denote them by Γ, and Γ .

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