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1 (Ryūta Hashimoto) α α p q < p/q α q Lagrange < <.5.4 < <.4.44 < < π Yes, I have a number Napier e

730508... 5.360979....36 0679 π Yes, I have a number.

2 ( ) > p q p q p q q q p q < p q, 3, 4 3, 7 5, 7, 4 7, 4 9, 99 70, 40 99, 39 69, , , ,

44356 > 69537 599448 599448 p q p q p q q q p q < p q, 3, 4 3, 7 5, 7, 4 7, 4 9,

3 , +, + +, + + +,..., 3, 7 5, 7, 4 9, 99 70, 39 69, , ,.... ξ [ξ [ξ [ξ ξ < [ξ + ξ ξ ξ ξ ξ ξ < ξ +, ξ < ξ ξ α α α c 0 α c 0 α α c α α c α 3 α 0 α α 0 α, c 0 α 0, α, c α, α, α 0 c 0 α c..., c k α k, α k+ α k c k,... c k k n c n α n 3

4 α k+ α k c k α k c k + α k+ α α 0 α c 0 + c 0 + α c + c 0 + c + α c + α 3 α [ c 0, α [ c0, c, α [ c0, c, c, α 3 [ 3,,,,,..., [ π 3, 7, 5,, 9,,,..., [ 5, 4, 4, 4, 4,..., [ e,,,,, 4,,.... α [ c 0, c,..., c k α k [ [ [,.44...,,.44...,,, [ [, 3, [,, 7 5 ( ) a b GL(, Z) α c d ( a c ) b α aα + b d cα + d ( ) 0 α α; B(Aα) (BA)α ( A, B GL(, Z) ) 0 4

5 ( ) c + α c α 0 α [ c 0, c,..., c n, α n c0 + [ c,..., c n, α n ( ) c 0 [ c,..., c n, α n 0 ( ) ( ) c 0 c [ c,..., c n, α n 0 0 ( ) ( ) ( ) c 0 c c n α n ( ) ( ) ( ) ( ) c 0 c c k c k {p k }{ } ( p k ) ( ) ( ) ( ) ( ) c 0 c c k c k ( ) ( ) ( ) p k p k c k 0 ( ) ( ) ( ) ( ) ( ) p k p k c 0 c c k c k () ( p k ) ( ) ( p k p k p k p k ) ( ) c k 0 ( c k p k + p k c k + p k ) p k c k p k + p k, p, p 0 c 0 ; c k +, q 0, q 0 () {p k }, { } p k / k ) ( ) ( ) [ (c 0 c c k c0, c,..., c k c k ( ) p k p k c k p k c k + p k p k c k + 5

0 ( ) ( ) ( ) p k p k c k 0 ( ) ( ) ( ) ( ) ( ) p k p k c 0 c c k c k () 0 0 0 0 ( p k ) ( ) ( p k p k p k p k ) ( ) c k 0 ( c k p k + p k c

6 () p k p k ( ) k+ (3) p k.3 (3) p k p k ( )k+ (4) k (k ) p k p k c k p k + p k c k + p k c k p k p k ( )k c k p k p k ( ) k c k p k p k ( )k c k (5) k (k ) (4) k p /q > p 0 /q 0 (4) k p /q < p /q p /q > p 0 /q 0 (5) k k 3 p /q < p 3 /q 3 < p /q p 0 q 0 < p q < < p k q k < p k+ q k+ < < p k+ q k+ < p k q k < < p 3 q 3 < p q α { } c k + q 0 q q 0 q < q < q 3 < α α [ ( p k c 0, c,..., c k, α k+ α p k α k+p k + p k α k+ + p k p k ) p k p k (α k+ + ) α k+ α k+p k + p k α k+ + ( ) k (α k+ + ) 6

7 { } α 0 α p k (α k+ + ) < (c k+ + ) + α p k (α k+ + ) > ((c k+ + ) + ) (+ + ) (+ + ) < α p k < (6) + α < α p k < (7) α α α α () α {c k } k 0 c 0 0 () {p k / } k {c k } k [ Euclid a, b Euclid a/b a, b ax by a/b 7

8 .5 ξ ξ α 0 α, c 0 α 0, α c 0, c α α, α 0 c, α... c k α k, α k+ c k,... α k ξ ξ {c k } {c k } ξ GL(, Z) ξ SL(, Z). p/q α p q p q q q α p q < α p p q p q q q qα p < q α p α 3 [ (443.B) [K q 8

9 α p q q qα p < q q α p α p q. α p/q p/q 3 p/q < p 0 /q 0 p/q > p /q p k / p k+ /+ p/q k α c 0 < qα p p/q 3 p k < α < p k+ < p + q < p k p k < p q < p k+ < α < p k + k k (8) k α p q > p k+ p + q qp k+ p+ q+ q+ 0 q (7) qα p > + > α p k (9) p q q p k < p k p k (8) 0 (3) 9

10 < q p/q < q qα p < α p k (9) p/q.3 α α > / 0 k 0 < q p/q qα p q gcd(p, q) qα p p/q (p, q) (p k, ) p/q gcd(p, q) p p j q q j j q j q {q } j k q 0 q k 0j α α > / p 0 /q 0 j k j < k < q j α p j α p k < + q j + q j+ + (p, q) 3 4 (7) {q } + < + {q } j < k j k.4 p k c k p k +p k c k + c kp k +p k c k + c k c k < c k c k k k k + α p k < α < p k+ + 0

11 k k + p k+ +p k + + p k < p k+ + p k + + < α < p k+ + Faray α p k < p k+ + p k + + < p k+ + p k + + < p k < p k+ + p k + + < p k+ + p k + + < α < p k+ + < (c k+ )p k+ + p k (c k+ )+ + < c k+p k+ + p k c k+ + + p k+ + < α < p k+ + [ [.5 Lagrange α k (6) α p k < < + q k p/q α p q < q α p q < q p/q α p q < q p/q Lagrange α p q < q p q α p/q q α p qα p q > q

12 q α p q q q α p q < q q q q q p q p q q q p q p q p q α + α p q < q q + q q + q q q 0 q < q + q q < q.6 Lagrange α p q < q p/q Legendre Lagrange Vahlen p k p k+ + α p q < q α p q < 5 q p/q Hurwitz Borel p k p k+ + p k+ + α p q < 5 q C α p q < C q C > 5 p/q C 5 α + 5 p/q C > + α 5 p/q C α + p/q C > 5 α + p/q

13 GL(, Z) [,,, [,,,... C 5,, 5 m,..., 9m Lagrange spectra 4 m Markoff 3 m + m + m 3mm m Markoff Markov Waldschmidt [W Markov Markoff (,, ) Markoff chain Markoff chain Markov 006 x + y + z xyz 0 < x y z x 3 + y 3 + z 3 xyz 0 < x y z x + y + z xyz 0 < x y z Markoff.7 Lagrange Pell Lagrange Pell D N N < D X DY N X, Y N ± Pell N N < D N [H,.5 [, 7 N > 0 0 < N (X + D Y )(X D Y ) 0 < X D Y D X Y < Y 0 < X D Y N X + D Y < N D Y < Y X/Y D 3

chain Markoff chain Markov 006 x + y + z xyz 0 < x y z x 3 + y 3 + z 3 xyz 0 < x y z x + y + z xyz 0 < x y z

14 N < 0 Y D X N D N > 0 < Y/X / D D Y X X X/Y D Lagrange 3 Pell [P [ 3. [,,,... [, 33 [ 7,,, 4,,,, 3,,, 3,,,, 4,,, 34 α [ c 0, c,..., c m, c m+,..., c m+l ( ) ( ) p m p m p m+l p m+l α α m+ α m+l+ q m q m q m+l q m+l α m+ α m+l+ α m+ α Lagrange {a k }{b k } α α 0 b 0 + D, a 0 (D b a 0); 0 b 0 + D c k, b k+ c k a k b k, a k+ D b k+ a 0 a k 3. D D [ D c0, c, c,..., c, c, c 0 4

15 α [ c, c,..., c n α α α [ c n,..., c, c 3.3 Pell D Pell X DY ± D l X DY X DY l X DY, X DY, X DY l (mod ), l 0 (mod 4), l (mod 4) N < D N X DY N N c ( ) k a k k c 4 Jacobi-Perron Padé Lehmar Lang ζ(3)rogers-ramanujan 5

16 [ (00) [H Hua, Loo Keng ( ). Introduction to Number Theory, Translated from Shu lun tao yin( ) by Peter Shiu, Springer-Verlag (98). [ 3 (985) [ (99) [K A. Ya. Khinchin. Continued Fractions. Dover(997). [ (987) [P O. Perron. Die Lehre von den Kettenbrüchen, Band I: Elementare Kettenbrüche, dritte, verbesserte und erweiterte auflage edition, B. G. Teubner Verlagsgesellschaft m. b. H., Stuttgart (954). [ (97) [W W. M. Waldschmidt. Open Diophantine Problems, Moscow Math. Journal vol. 4, no. (004), pp hasimoto@dg.takuma-ct.ac.jp Ryūta Hashimoto Takuma National College of Technology Takuma-cho, Mitoyo-shi, Kagawa JAPAN hasimoto/ 6

Teubner Verlagsgesellschaft m. b. H., Stuttgart (954). [ (97) [W W. M. Waldschmidt. Open Diophantine Problems, Moscow Math. Journal vol. 4, no. (004), pp. 45-305. http://arxiv.

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6 26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7