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1 1. / / 2. R 4k / 7. /n 8. n 1 / / Z d ( R d ) d P Z d R d R d? n > 0 n 1.1. R 2 6 n 5 n [Scherrer 1946] d 3 R (Schoenberg 1937). d 3 R d n n = 3, 4, d 3 R d 1.3. θ θ/π < θ < π/2 cos θ θ = π/3 θ/π [2, 11] 1.5 ( G. Pick 1899). I B I + B/2 1 1

2 I + B = 4 I + B = 3 1/2 2 R 4k 2.1 (Lagrange 4 ). [5, 12] 2.2 (M 1995). n = 2 n = 4k (k 1) P Z n R n f : R n R n { f(o) = O, f(p ) = (, 0,..., 0) f(z n ) Z n.. (a, b) (0, 0) (x, y, z, w) (0, 0, 0, 0) ( ) x y z w a b [[a, b]] =, [[x, y, z, w]] = y x w z b a z w x y w z y x R 2, R 4 Z 2 P = (a, b) (, 0), Z 4 (x, y, z, w) (, 0, 0, 0) ( ) [[x, y, z, w]] 0 n = 8 P = (..., x, y, z, w) 8 8 (p, q, r, s, m, 0, 0, 0) ( 0 ) [[x, y, z, w]] [[p, q, r, s]] mi 8 8 mi [[p, q, r, s]] t (, 0,..., 0) R 8 n = M (M M t = λi) P (, 0..., 0) λ ( ) = a 2 + b 2 + c 2 + d 2 a, b, c, d Z [[a, b, c, d]] 0 P (,,,,, 0,..., 0) 0 M (, 0..., 0) f < n 0 (mod 4) X Z 4k R 4k (k 1) R 4k X Z 4k 1 R 4k 1. X P P 2.2 f X 2.1. R 4 R n = 4k 1 R n n 2.2. R n n [Schoenberg 1937] n n+1 n 1 (mod 4) n + 1 n 3 (mod 4) 2

3 m 0 8m + 7 = b2 +c 2 +d 2 a 2 a, b, c, d Z. a, b, c, d a, b, c, d a b n n 2 0, 1, 4 (mod 8) a a 2 (8m + 7) 7, b 2 + c 2 + d 2 7 (mod 8) a a 2 (8m + 7) 0 (mod 4), b 2 + c 2 + d 2 0 (mod 4) 3.1. a 2 (8m + 7) 3.1 (Legendre ). n n 4 i (8m + 7) (i, m 0) A, B, C Z n ABC Z n Θ n Θ n (Beeson 1992). 1. θ Θ 2 θ = π/2 or tan θ Q Θ n = {θ θ = ABC, A, B, C Z n } 2. θ Θ 4 θ = π/2 or tan 2 θ = a2 +b 2 +c 2 d 2 (a, b, c, d Z) 3. θ Θ 5 cos 2 θ Q 4. Θ 2 Θ 3 = Θ 4 Θ 5 = Θ 6 = θ Θ n cos 2 θ Q. cos 2 θ Q tan 2 θ Q θ Θ 5 4. π/3 Θ 3 \Θ 2, arctan 7 Θ 5 \Θ Θ 3 = Θ 4. θ Θ n cos 2 θ Q θ Θ 5. 4 T T 1, 7, T T θ Θ 5 cos 2 θ Q Θ 5 3

4 4.1. v 1,..., v k a ij = v i v j det(a ij ) v 1,..., v k v 1,..., v k Gram 4.2. S = A 1... A k F S A i F 2 Q (i = 1..., k) F. A 1 F = x 2 A 1 A x k A 1 A k A 1 A i A 1 F x 2 A 1 A 2 A 1 A i + + x k A 1 A k A 1 A i = A 1 F A 1 A i (i = 2,..., k) (1) A 1 F A 1 A i = ( A 1 F 2 + A 1 A i 2 F A i 2 )/2 Q A 1 A i A 1 A j Q (1) (1) x 2,..., x k Q F 4.1. T d d 1 d < k d = k T = A 0 A 1... A k T R n+4 (n > k) A 1,..., A k R n {(0, 0, 0, 0)} A 0 A i 2 Q (i = 1, 2,..., k) A 0 S := A 1... A k F h = A 0 F T T 2 S (k 1) S 2 T = 1 k S h h2 = A 0 F 2 Q i A i F 2 Q 4.2 F Q n {(0, 0, 0, 0)} T F = (0,..., 0) Q n+4 h 2 h 2 = a2 +b 2 +c 2 +d 2 e a, b, c, d, e Z A 2 0 = (0,..., 0, a e, b e, c e, d e ) Qn+4 A 0, A 1,..., A k T T 4.2 (M 1995). X (1) X ( AB 2 (2) A, B, C X BC ) Q. (3) A, B, C X ABC Θ 5.. (1) (2) (3) (3) (1): X k k 2 X k T T X (1) (2) (2) X A, B, C AB BC Q 4.1. Y Y R ABC Θ 5 A = α π 2 4

5 (1) ABC R 5 (2) α Θ 3 ABC R 3 (3) α Θ 2 ABC R k = 2, 3 π 2 Θ k Θ k 5 T Z d T Z d? n n Z d d δ(n) 4.2 δ(n) n ( 1998). n T T Q n+3 ( R n+3 ) n Lagrange a, b, c, d, e ax 2 = by 2 + cz 2 + du 2 + ev 2 (x, y, z, u, v) [Meyer 1884] rank 5 R Q T = A 0 A 1 A 2... A n v i := A 0 A i (i = 1, 2,..., n) Schmidt u 1 = v 1, u 2 = v 2 v 2 u 1 u 1 u 1 u 1, u 3 = v 3 v 3 u 1 u 1 u 1 u 1 v 3 u 2 u 2 u 2 u 2,.... B 0 = A 0, B 0 B i = u i (i = 1, 2,..., n) B 0 B 1 B 2... B n A T Z d T Z d 5.2. Θ 4 5

6 . ξ, η, ζ (ξη) 2 (ξ/η) 2 = (ξη) 2 /η 4 ξ, η Θ 4 (ξη) 2, (ηζ) 2 4 i (8k + 7) (mod 8) (ξη 2 ζ) 2 (ξ/ζ) 2 (b 2 + c 2 + d 2 )/a 2 (a, b, c, d Z) ξ, ζ Θ Θ Θ (M 1998). Z 5. OABC OA, OB, OC 5.2 OAB, OBC, OCA Z 4 O, A, B Z 4, O = (0, 0, 0, 0) OC = η η 2 = x 2 +y 2 +z 2 +w 2 x, y, z, w Z [[x, y, z, w]] OAB η OA B O, A, B Z 4 Z 4 {0} Z 5 C = (0, 0, 0, 0, η 2 ) OA B C OABC Z 5 δ(1) = 1, δ(2) = δ(3) = n 3 δ(n) n Z 4 Z 3 Θ 3 Z 4 6 / f : S S f f = id f S (involution) 6.1. S f : S S [f ] S mod 2. { n 6.1. n N n = n {n } a n/a {n } 6.1 ( ). p = 4k + 1 p = a 2 + b 2 (a, b N). D. Zagier 5 = = = 9 k > 2 S = {(x, y, z) N 3 x 2 + 4yz = p} A = {(x, y, z) S x < y z} B = {(x, y, z) S y z < x < 2y} C = {(x, y, z) S 2y < x} 6

7 < = (1, k, 1) A, (1, 1, k) B, (3, 1, k 2) C A, B, C S = A B C(disjoint union) A (x, y, z) (x + 2z, z, y x z) C A = C B (x, y, z) (2y x, y, x y + z) B B (1, 1, k) B S S S (x, y, z) (x, z, y) S (x, y, y) p = x 2 + 4y 2 = x 2 + (2y) 2 x, y N 7 /n? 1957) n > 0 n 7.1 (Sierpinski). ( 2, 1 3 ) R2 n? n N n 7.2. p 1 (mod 4) w w = p k w Z[i] 4(k + 1). p = a 2 + b 2 p = (a + bi)(a bi) w w = p k = (a + bi) k (a bi) k a + bi, a bi Z[i] w u(a + bi) s (a bi) k s (s = 0, 1, 2,... k, u = ±1, ±i) w w = p k w Z[i] 4(k + 1) 7.1 (M+ 1998). p p 1(mod 4), p k 1 (mod 8) (4x 1) 2 + (4y) 2 = p k k X 2 + Y 2 = p k (X, Y ) 4(k + 1) ( ) 2 0, 1, 4 (mod 8) X 2 + Y 2 1 (mod 8) X 2 1, Y 2 0 (mod 8) X 2 0, Y 2 1 (mod 8) X 2 + Y 2 = p k X ±1; Y 0 (mod 4) X 0; Y ±1 (mod 4) (4x 1) 2 + (4y) 2 = p k (x, y) X 2 + Y 2 = p k, X 1 (mod 4) (2) (X, Y ) X 2 +Y 2 = p k (±A, B), (B, ±A) (B 0 mod 4) 1 (2) (4x 1) 2 + (4y) 2 = p k 4(k + 1)/4 = k n n 7.2. p 1 (mod 4) (2x 1) 2 + (2y) 2 = p k 2k n = 3, 4, 5,..., 10 n 7.1, 7.2 7

8 8 n 8.1. R 3 (4x 1) 2 + (4y) 2 + (4z 2) 2 = 17 k + 2 k + 1 z = 0? n 4 R 3 ( ) { n ( ) 8.1 (M 2006). n > d 2 n R d n d 8.1. M 1, M 2,..., M s a 1, a 2,..., a s Z s a i Z a i Z, i = 1, 2,..., s. M i M i d = 5 k 0 + k 1 + k 2 + k = n, k 0 2 k i p 0, p 1, p 2, p 3, M 1, M 2, M 3 p 0 p 1 p 2 p 3 1 (mod 8) p k pk pk 3 3 < p k0 0 < M 1 < M 2 < M 3 Dirichlet τ i = 2M i + pk i i p k 0 0 (i = 1, 2, 3) R 5 (x, y, z 1, z 2, z 3 ) 8M i Σ (4x 1) 2 + (4y) 2 + (4z 1 τ 1 ) 2 + (4z 2 τ 2 ) 2 + (4z 3 τ 3 ) 2 = p k τ τ τ 2 3. n (x, y, j 1, j 2, j 3 ) Σ (4x 1) 2 + (4y) 2 = p k τi 2 (4j i τ i ) 2 = p k 0 0 (16ji 2 8j i τ i ) = p k (ji 2 j i M i ) + j i (p k i i p k 0 0 ) M i j i = s i M i (s i Z) (4x 1) 2 + (4y) 2 = p k M i 2(s2 i s i) + 3 s i(p k i i p k 0 0 ) s i = 0, 1 (4x 1) 2 + (4y) 2 = p k0 0 + s i (p ki i p k0 0 ) = pk0 < 2p k 0 0 (s 1 + s 2 + s 3 )p k s i p ki i (s 1 + s 2 + s 3 )p k0 0 s 1 + s 2 + s 3 1 (s 1, s 2, s 3 ) (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) (4x 1) 2 + (4y) 2 = p k i i (i = 0, 1, 2, 3) k i + 1 Σ n n (,, 0, 0, 0), (,, 0, 0, 0), (,, 0, 0, 0), (,, M 1, 0, 0), (,, 0, M 2, 0), (,, 0, 0, M 3 ) 5 8

9 8.1. n 4 R 3 ( ) 8.1. n 4 R 3 n n [1] M. J. Beeson, Triangles with vertices on lattice points, Amer. Math. Monthly 99(1992) [2] P. Frankl [3] W. W. Funkenbusch, From Euler s formula to Pick s formula using an edge theorem, Amer. Math. Monthly, 81(1974) [4] H. Hadwiger and H. Debrunner (Translated by Victor Klee), Combinatorial Geometry in the Plane, Holt, Rinehart and Winston, New York [5] A. Y. Khinchin [6] T. Kumada, Isometric embedding of metric Q-vector space into Q N, Europ. J. Combin. 19(1998) [7] H. Maehara, Embedding a polytope in a lattice, Discrete Comput Geom 13(1995) [8] H. Maehara, Embedding a set of rational points in lower dimensions, Discrete Math. 192(1998) [9] H. Maehara and M. Matsumoto, Is there a circle that passes through a given number of lattice points?, European J. Combin. 19(1998) [10] H. Maehara, On a sphere that passes through n lattice points, to appear. [11] I. Niven, Numbers: Rational and Irrational, MAA, [12] I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, Wiley, New York [13] W. Scherrer, Die Einlangerung eines regularen Vielecks in ein Gitter, Elemente der Math. 1(1946) [14] I. J. Schoenberg, Regular simplices and quadratic forms, J. London Math. Soc. 12(1937) [15] I. J [16] H [17] H. Steinhaus, One Hundred Problems in Elementary Mathematics, Dover Publications, Inc. New York [18] [19] D. Zagier, A one sentence proof that every prime p 1 (mod 4) is a sum of two squares, Amer. Math. Monthly 97(1990),

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