γ n ϑ n

n ψ T 8 Q 6

j, k, m, n, p, r, r t, x, y f m (x) (f(x)) m / a/b (f g)(x) = f(g(x)) n f f n I J α β I = α + βj N, Z, Q ϕ Εὐκλείδης ὁ Ἀλεξανδρεύς Στοιχεῖα ἄκρος καὶ μέσος

λόγος ὕδωρ αἰθήρ ϕ φ Φ τ ϕ Φειδίας τ τομή g G ϕ

,, 2, 3, 5,... /, 2/, 3/2, 5/3,... ϕ Υπατία ἡ Ἀλεξανδρεῖα ΑΓΕΟΜΕΤΡΗΤΩΣ ΜΗΔΕΙΣ ΕΙΣΙΤΩ

Ἀπολλώνιος ὁ Περγαῖος

qetir hgrannik, xestigrannik (kub), vosьmigrannik, dvenadcatigrannik, dvadcatigrannik Z AB AB : AZ = AZ : ZB a = AZ b = ZB

a > b AB = a + b a + b a = a b. a b = + a. b ϕ = a/b ϕ = + ϕ. ϕ 2 = ϕ +.

ϕ ξ 2 ξ = 0. ξ = + 5 2, ξ 2 = 5. 2 ϕ = + 5 2 =, 68033988... ϕ ϕ ϕ = + ϕ = + + ϕ = +. + + + ϕ = [; ], ϕ ϕ /ϕ ϕ = [0; ]. ϕ Ω

zolotoe seqenie ϕ ϕ ϕ ϕ = m/n m n m/n = (m + n)/n (m + n)/n m = m + n m = n m = 2m ϕ n a b a > b na + b = a a b.

a b = n + a. b n n n σ(n) = a/b σ(n) = n + σ(n). σ 2 (n) = nσ(n) +. n σ(n) λ 2 nλ = 0. λ = λ (n) = n + n 2 + 4 2, λ 2 = λ 2 (n) = n n 2 + 4. 2

n γ n : (0, ) (0, ) γ n (x) = n + x, ξ n = σ(n) y = γ n (x) y = x ξ n = σ(n) x = γ n (x) x (n) 0, γ n (x (n) 0 ), γ 2 n (x (n) x (n) 0 > 0 0 ), γ 3 n (x (n) 0 ),..., σ(n) = n + D(n), 2 D(n) = n 2 + 4 n 2 < D(n) < (n + ) 2,

γ n n < σ(n) < n +. σ(n) n n + ( n + ) D(n) (σ(n) n) = n = 0, n n 2 n n σ(n) σ(n) = n, {σ(n)} = σ(n). u u u u {u} {u} = u u u = u + {u}, 0 {u} <, u u < u + 3 = 3, 3.4 = 3, 4 = 4, 3.4 = 4. {3} = 0, {3.4} = 0.4, { 4} = 0, { 3.4} = 0.86. σ(n) n D(n) = n 2 + 4 m

n 2 + 4 = m 2 n m Πυθαγόρας ὁ Σάμιος c a b c 2 = a 2 + b 2 n σ(n) = n + σ(n) = n + n + σ(n) σ(n) = [n; n], = n +. n + n + n + n σ(n)

ϕ = σ() ψ = σ(2) χ = σ(3) n n n n D(n) σ(n) + 5 2 ϕ, τ + 2 ψ 3+ 3 2 χ 2 + 5 5+ 29 2 3 + 0 σ(n) = + nσ(n). σ(n) = + nσ(n) = + n + nσ(n) = + n + n + n +.... n ϑ n : (0, ) (0, ) ϑ n (x) = + nx, ξ n = σ(n) y = ϑ n (x) y = x ξ n = σ(n) x = ϑ n (x) x (n) 0, ϑ n (x (n) 0 ), ϑ 2 n (x (n) 0 ), ϑ 3 n (x (n) 0 ),...,

ϑ n x (n) 0 > 0 n Q n = {α + βσ(n); α Q, β Q} Q Q n (α + βσ(n)) + (α + β σ(n)) = (α + α ) + (β + β )σ(n), (α + βσ(n)) (α + β σ(n)) = (αα + ββ ) + (nββ + αβ + α β)σ(n), Q n 0 + 0 σ(n) = 0 + 0 σ(n) = α 0 β 0 α + βσ(n) = α + nβ α 2 + nαβ β 2 β α 2 + nαβ β 2 σ(n). α 2 + nαβ β 2 α/β = ( n ± D(n))/2 D(n) α/β Q n α + βσ(n) = α + β σ(n) α = α β = β.

F m (x) m x F m+2 (x) = xf m+ (x) + F m (x) (m = 0,, 2,...) F 0 (x) = 0, F (x) = F 0 (x) = 0, F (x) =, F 2 (x) = x, F 3 (x) = x 2 +, F 4 (x) = x 3 + 2x, F 5 (x) = x 4 + 3x 2 +, F 6 (x) = x 5 + 4x 3 + 3x, F 7 (x) = x 6 + 5x 4 + 6x 2 +, F 8 (x) = x 7 + 6x 5 + 0x 3 + 4x, F 9 (x) = x 8 + 7x 6 + 5x 4 + 0x 2 +, F 0 (x) = x 9 + 8x 7 + 2x 5 + 20x 3 + 5x. F m (x) m m m F m (x) m L m (x) m x L m+2 (x) = xl m+ (x) + L m (x) (m = 0,, 2,...), L 0 (x) = 2, L (x) = x x = F m = F m (), L m = L m ().

L 0 (x) = 2, L (x) = x, L 2 (x) = x 2 + 2, L 3 (x) = x 3 + 3x, L 4 (x) = x 4 + 4x 2 + 2, L 5 (x) = x 5 + 5x 3 + 5x, L 6 (x) = x 6 + 6x 4 + 9x 2 + 2, L 7 (x) = x 7 + 7x 5 + 4x 3 + 7x, L 8 (x) = x 8 + 8x 6 + 20x 4 + 6x 2 + 2, L 9 (x) = x 9 + 9x 7 + 27x 5 + 30x 3 + 9x. L 0 (x) = x 0 + 0x 8 + 35x 6 + 50x 4 + 25x 2 + 2. L m (x) m 0 m m L m (x) m

x u m+2 xu m+ u m = 0, (m = 0,, 2,...).

u m = λ m, λ m+2 xλ m+ λ m = 0 λ 2 xλ = 0. λ = λ (x) = x + x 2 + 4 2 x, λ 2 = λ 2 (x) = x x 2 + 4. 2 λ + λ 2 = λ (x) + λ 2 (x) = x, λ λ 2 = λ (x)λ 2 (x) =. x λ (x) x λ 2 (x) u m = C λ m + C 2 λ m 2,

C C 2 m x C C 2 F m (x) = λm λ m 2 λ λ 2 = C + C 2 = 0, C λ + C 2 λ 2 =. C = λ λ 2 = x2 + 4 = C 2. 2 m x 2 + 4 ((x + x 2 + 4) m (x x 2 + 4) m ). C C 2 C + C 2 = 2, C λ + C 2 λ 2 = x. C = C 2 =. L m (x) = λ m + λ m 2 = 2 m ((x + x 2 + 4) m + (x x 2 + 4) m ). F m (x) = L m (x) = 2 m j 0 2 m j 0 ( ) m (x 2 + 4) j x m 2j, 2j + ( ) m (x 2 + 4) j x m 2j, 2j F(x, t) = F m (x)t m m=0

L(x, t) = L m (x)t m. m=0 F(x, t) = λ λ 2 L(x, t) = (λ m λ m 2 )t m = λ λ 2 m=0 = t xt t 2. (λ m + λ m 2 )t m = m=0 = 2 xt xt t 2. ( ) λ t = λ 2 t λ t + λ 2 t = x t < /λ = x 2 + 4 x F(x, t) = t 2 xt, L(x, t) = xt t2 xt t. 2

F(0, t) = t t = 2 m=0 t 2m+ = F m (0)t m m=0 F m (0) = m F m (0) = 0 L(0, t) = 2 t 2 = 2 m=0 t 2m = L m (0)t m, m=0 L m (0) = 0 m L m (0) = 2 L x (x, t) = t(t2 + ) ( xt t 2 ) = 2 F t (x, t) = t 2 + ( xt t 2 ) 2 = m= m= L m(x)t m, mf m (x)t m = t L m(x) = mf m (x) mf m (x)t m. m λ (x) = 2 (x + x 2 + 4) = ( ) x + = λ (x) 2 x2 + 4 x2 + 4 m= λ 2(x) = 2 (x x 2 + 4) = 2 ( ) x = λ 2(x) x2 + 4 x2 + 4. L m(x) = (λ m (x) + λ m 2 (x)) = m(λ m (x)λ (x) + λ m 2 (x)λ 2(x)) =

= m x2 + 4 (λm (x) λ m 2 (x)) = mf m (x). (x 2 + 4)F m(x) = ml m (x) xf m (x). (x 2 + 4)F m(x) + 3xF m(x) (m 2 )F m (x) = 0. F m (x) (x 2 + 4)y + 3xy (m 2 )y = 0 y(0) = ( ( ) m )/2 y (0) = m( + ( ) m )/2 m m > 0 F m (x) = ( ) m+ F m (x), L m (x) = ( ) m L m (x). m m < 0 F m (x) = = λ λ 2 ( ( λ 2 ) m ( λ ) m) = λ λ 2 ( ) m (λ m 2 λ m ) = ( ) m+ F m (x) L m (x) = ( λ 2 ) m + ( λ ) m = ( ) m (λ m 2 + λ m ) = ( ) m L m (x). {F m (x)} m Z {L m (x)} m Z m = 0 m :... 3 2 0 2 3... F m (x) :... x 2 + x 0 x x 2 +... L m (x) :... x 3 3x x 2 + 2 x 2 x x 2 + 2 x 3 + 3x...

(x 2 + 4)F 2 m(x) = L 2 m(x) + 4( ) m+. (x 2 + 4)F 2 m(x) = (λ m λ m 2 ) 2 = λ 2m + λ 2m 2 2(λ λ 2 ) m = = λ 2m + λ 2m 2 + 2( ) m + 4( ) m+ = (λ m + λ m 2 ) 2 + 4( ) m+. λ m,2(x) = F m (x)λ,2 (x) + F m (x). F m (x)λ (x) + F m (x) = λ λ 2 ( (λ m ) λ m 2 )λ + λ m λ m 2 = = ( ) λ m+ + λ m 2 + λ m λ m 2 = (λ m+ λ m λ 2 ) = λ λ 2 λ λ 2 = λ λ 2 λ m (λ λ 2 ) = λ m. λ m + λ m 2 = F m (x)λ + F m (x) + F m (x)λ 2 + F m (x) = = F m (x)(λ + λ 2 ) + F m (x) + F m (x) = xf m (x) + F m (x) + F m (x) = = F m (x) + F m+ (x). L m (x) = F m (x) + F m+ (x). L m (x) (x 2 + 4)y + xy m 2 y = 0

y(0) = +( ) m y (0) = m( ( ) m )/2 2m + y(0) = 0 y (0) = 2m + 2m + y = a k (m)x k, k=0 a 0 (m) = 0 a (m) = 2m+ k > 0 a k+2 (m) = (2m + )2 k 2 4(k + )(k + 2) a k(m). a 2k = 0 k a 2k+ = 0 k > m a 3 (m) = (2m + )m(m + ), a 5 (m) = 3 2! a 2k+ (m) = 2m + ( ) m + k. 2k + 2k (2m + )(m )m(m + )(m + 2). 5 4!

m L 2m+ (x) = k=0 2m + 2k + ( m + k 2k ) x 2k+. m ( ) m m + k L 2m (x) = 2 + x 2k. k 2k k= L m (x) = m 2 k=0 m m 2k ( m k k ) x m 2k + + ( ) m. F m (x) = L m(x)/m m > 0 F m (x) = m 2 k=0 ( m k k ) x m 2k. m 2 ( ) m k F m =. k k=0 (x + ) m x x = ex e x, x = ex + e x. 2 2

2 x 2 x =.

x + x = e x, x x = e x. [0, ) [, ) x = (x + x 2 + ). π y = a(x)y 2 + b(x)y + c(x)

x 2 + y 2 = x 2 y 2 = x x = 2 t t = (x/2) t λ = t + t = e t, λ 2 = t t = e t. L m (x) = λ m + λ m 2 = e mt + ( ) m e mt. L 2k (x) = 2 (2kt) = 2 (2k (x/2)), L 2k+ (x) = 2 ((2k + )t) = 2 ((2k + ) (x/2)).

m L m (x) = 2 (m (x/2)), m L m (x) = 2 (m (x/2)), m n (L n L m )(x) = L n (L m (x)) = L mn (x). m n n m mn L n (L m (x)) = 2 (n (L m (x)/2)) = 2 (n ((m (x/2)))) = = 2 (nm (x/2)) = L mn (x). n m mn L n (L m (x)) = 2 (n (L m (x)/2)) = 2 (n ((m (x/2)))) =

= 2 (nm (x/2)) = L mn (x). m n L n (L m (x)) = L m (L n (x)) = L mn (x). λ λ 2 m λ m + λ m 2 = L m (x), λ m λ m 2 =. µ = λ m µ 2 = λ m 2 µ 2 L m (x)µ = 0. n ν = µ n = λ mn ν 2 = µ n 2 = λ mn 2 ν 2 L mn (x)ν = 0 ν 2 L n (L m (x))ν = 0. m, m 2,..., m r λ λ 2 L (x)λ = 0, µ = λ m m 2 m r µ 2 L m m 2 m r (x)µ = 0. L m m 2 m r (x) = (L m L m... L mr )(x).

L m (x) (x 2 + 4)y + xy m 2 y = 0. m y y 2 y = C y + C 2 y 2 x t x = 2 t x 2 + 4 = 4 2 t + 4 = 4 2 t y = dy dx = dy dt dt dx = ẏ 2 t, y = dy dx = ÿ 4 2 t ẏ t 4 3 t. t ÿ m 2 y = 0. y = C (mt) + C 2 (mt),

y = C (m (x/2)) + C 2 (m (x/2)). F m (x) (x 2 + 4)y + 3xy (m 2 )y = 0. (x 2 + 4)y + 2xy + xy + y m 2 y = (x 2 + 4)y + 3xy (m 2 )y = 0. z = y (x 2 + 4)z + 3xz (m 2 )z = 0. ( x) = x, ( x) = x, ( x) = z = x2 +, x2 + 4 (C (m (x/2)) + C 2 (m (x/2))). C C 2 y = L m (x) y(0) = + ( ) m ) y (0) = m( ( ) m )/2 z = F m (x) z(0) = ( ( ) m )/2 z (0) = m( + ( ) m )/2 mf m (x) = L m(x) z,m (x) = y,m (x) = (n (x/2)), y 2,m (x) = (m (x/2)), x2 + 4 (m (x/2)), z 2,m(x) = (m (x/2)) x2 + 4

M(x) = 0 x,

x M(x) λ x λ = λ2 λx. M(x) m m M m (x) = 0 = a m(x) b m (x) x c m (x) d m (x), a m (x), b m (x), c m (x), d m (x) x a 0 (x) =, b 0 (x) = 0, c 0 (x) = 0, d 0 (x) =, a (x) = 0, b (x) =, c (x) =, d (x) = x. M m+ (x) = M m (x)m(x) a m+(x) c m+ (x) b m+ (x) d m+ (x) = a m(x) c m (x) b m (x) d m (x) 0 x. a m+ (x) = b m (x), b m+ (x) = a m (x) + xb m (x), c m+ (x) = d m (x), d m+ (x) = c m (x) + xd m (x). b m+2 (x) = xb m+ (x) + b m (x), d m+2 (x) = xd m+ (x) + d m (x).

a m = F m (x), b m (x) = F m (x), c m (x) = F m (x), d m (x) = F m+ (x). 0 x m = F m (x) F m (x) F m (x) F m+ (x). m F m (x)f m+ (x) F 2 m(x) = ( ) m. F m r (x)f m+r (x) F 2 m(x) = ( ) m+ r F r (x), m r

F m (x)f r+ (x) F m+ (x)f r (x) = ( ) m F m r (x), M p (x)m q (x) = M p+q (x) p q

F p (x) F p (x) F p (x) F p+ (x) F q (x) F q (x) F q (x) F q+ (x) = F p+q (x) F p+q (x) F p+q (x) F p+q+ (x). p = q = m p = m + q = m F p+q (x) = F p (x)f q (x) + F p (x)f q+ (x). F 2m (x) = F m (x)l m (x), F 2m+ (x) = F 2 m(x) + F 2 m+(x). x = F 2m = F m L m, F 2m+ = F 2 m + F 2 m+. m F m (x) F m/2 (x) m F m F m/2 σ(n) λ 2 nλ = 0 σ(n) = λ (n), λ (x) σ m (n) m n Q n

σ m (n) = F m (n) + F m (n)σ(n). σ 0 (n) =, σ (n) = σ(n), σ 2 (n) = + nσ(n), σ (n) = n + σ(n). σ(n) m m > 0 σ m (n) = (σ (n)) m σ 2 (n) = nσ(n)+ σ m+2 (n) = nσ m+ (n) + σ m (n). n \ m F m (n) n = D() = 5 λ = ( + 5)/2 λ 2 = ( 5)/2 F m () = (( 2 m + 5) m ( ) 5) m, 5 m F m F m (n) m n F m+2 (n) = nf m+ (n)+f m (n) F 0 (n) = 0, F (n) = n = 2 D(2) = 8 λ = + 2 λ 2 = 2 F m (2) = (( 2 + ) m ( 2 ) m ) 2, 2

m P m P m = F m (2) m 2 P m+2 = 2P m+ + P m P 0 = 0, P = n = 3 D(3) = 3 λ = (3 + 3)/2 λ 2 = (3 3)/2 F m (3) = 2 m 3 (( 3 + ) m ( 3 3 ) m ) 3, m H m H m = F m (3) m 3 H m+2 = 3H m+ + H m H 0 = 0, H = L m (n) m n L m+2 (n) = nl m+ (n) + L m (n) L 0 (n) = 2, L (n) = n n = L m () = 2 m ( ( + 5) m + ( 5) m ). L m (2) = (( + ) m ( 2 + ) m ) 2. L m (3) = (( 3 + ) m ( 3 + 3 ) m ) 3. 2 m σ σ 4 D() = 5 σ(4) σ() σ(4) = + 2σ() = ϕ 3. σ m (n) σ(m) M n n = m = 3

M = 4 D() = 5, D(4) = 20 = 4 5 = 4D() σ() σ(4) Q σ(4) ϕ = σ() σ() σ(4) = (4 + 20)/2 = 2 + 5 = 2 + (2ϕ ) = + 2ϕ = ϕ 3. n = M = D(M)/D(n) > n = M = D()/D() = 25/5 = 25 σ() = ( + 25)/2 = ( + 5 5)/2 = ( + 5(2ϕ ))/2 = 3 + 5ϕ = ϕ 5. x = n M = L 2m+ (n) (n 2 + 4)F 2 2m+(n) = L 2 2m+(n) + 4. σ(m) = (M + M 2 + 4)/2 = (M + F 2m+ (n) n 2 + 4)/2 = = (M + F 2m+ (n)(2σ(n) n))/2 = (M nf 2m+ (n))/2 + F 2m+ (n)σ(n). M nf 2m+ (n) = L 2m+ (n) nf 2m+ (n) M nf 2m+ (n) = F 2m + F 2m+2 nf 2m+ (n) = 2F 2m (n). σ(l 2m+ (n)) = F 2m (n) + F 2m+ (n)σ(n) = σ 2m+ (n). µ = λ 2m+ (n) = σ 2m+ (n) µ 2 L 2n+ (n)λ = 0,

λ (n) = σ(n) λ 2 nλ = 0. σ(l 2n+ (n)) = σ 2m+ (n). M ξ 2 Mξ = 0 M ξ 2 Mξ + = 0. [n; n] 2m+ = [L 2n+ (n); L 2n+ (n)]. m F m () L m () F m (2) L m (2) F m (3) L m (3) F m (4) L m (4) F m (n) L m (n) σ(4) = + 2ϕ = ϕ 3,

σ() = 3 + 5ϕ = ϕ 5, σ(29) = 8 + 3ϕ = ϕ 7, σ(76) = 2 + 34ϕ = ϕ 9, σ(99) = 55 + 89ϕ = ϕ, σ(4) = 2 + 5ψ = ψ 3, σ(82) = 2 + 29ψ = ψ 5, σ(478) = 70 + 69ψ = ψ 7, σ(2786) = 408 + 985ψ = ψ 9, σ(6258) = 2378 + 574ψ = ψ, σ(36) = 3 + 0χ = χ 3, σ(393) = 33 + 09χ = χ 5, σ(4287) = 360 + 89χ = χ 7, σ(46764) = 3927 + 2970χ = χ 9, σ(507) = 42837 + 448χ = χ. n M M n σ(m) σ(n) σ(m) = a + bσ(n), a b σ(m) Q n k D(M) = k 2 D(n) M 2 + 4 = k 2 (n 2 + 4). σ(m) = M + D(M) 2 = M + k(2σ(n) n) 2 = M + k D(n) 2 = M nk 2 = + kσ(n).

a = M nk, b = k. 2 n M 2 + 4 = k 2 (n 2 + 4) M 2 + 4 n 2 + 4 M 2 + 4 > n 2 + 4 k > L 2 2m+(n) + 4 = (n 2 + 4)F 2 2m+(n), p, q, r M = L 2p+ (r) n = L 2q+ (r) M 2 + 4 n 2 + 4 = L2 2p+(r) + 4 L 2 2q+(r) + 4 = (r2 + 4)F 2 2p+(r) (r 2 + 4)F 2 2q+(r) = k = F 2p+ (r)/f 2q+ (r) σ(m) = M nk 2 + kσ(n). ( ) 2 F2p+ (r). F 2q+ (r) σ(m) σ(n) r 2p +, 2q + M = L 2p+ (r) n = L 2q+ (r) M = = L 5 (), n = 4 = L 3 () k = F 5 ()/F 3 () = 5/2 σ() = 4 (5/2) 2 + 5 2 σ(4) = 2 + 5 2 σ(4). σ() = + 2 + 4 2 σ() = 2 = 2 + 5 5 2, σ(4) = 4 + + 5(σ(4) 2) 2 6 + 4 2 = 2 + 5 2 σ(4). = 2 + 5.

M N σ(m) σ(n) σ(n) σ(m) σ(n) σ(n) σ(m) = α + βσ(n), σ(n) = α + β σ(n). σ(m) = α + β σ(n), α, β, α, β, α, β σ(76) = 2 + 34ϕ σ() = 3 + 5ϕ 5σ(76) 34σ() = 05 02 = 3 σ(76) = 3 5 + 34 5 σ(). p q r 2 (r + r 2 + 4) p + p 2 (r r 2 + 4) p = L p p (r), 2 (r + r 2 + 4) p p 2 (r r 2 + 4) p = r 2 + 4F p p (r). 2σ p (r) = L p (r) + r 2 + 4F p (r). 2σ q (r) = L q (r) + r 2 + 4F q (r). r2 + 4 = 2σp (r) L p (r) F p (r) σ p (r) = F q(r)l p (r) F p (r)l q (r) 2F q (r) = 2σq (r) L q (r). F q (r) + F p(r) F q (r) σq (r).

σ 3 (3) = 33 + 0 33 σ4 (3), σ 4 (3) = 0 + 33 0 σ3 (3). k=0 ( ) k n. σ(n) q = n/σ(n) < k=0 ( ) k n = σ(n) n σ(n) k= = σ 2 (n) σ 2 (n) nσ(n) = σ2 (n). ( ) k n = σ 2 (n) = nσ(n). σ(n) k=0 ( ) k n = σ 2 (n), σ(n) k= ( ) k n = nσ(n). σ(n) ϕ 2 = + ϕ σ(n) λ 2 nλ = 0 σ(n) = λ (n) λ 2 (n) m µ = λ 2m (n), µ 2 = λ 2m 2 (n). µ + µ 2 = λ 2m (n) + λ 2m 2 (n) 2 = L 2m (n) 2

µ µ 2 = (λ 2m (n) )(λ 2m 2 (n) ) = = (λ (n)λ 2 (n)) 2m (λ 2m (n) + λ 2m 2 (n)) + = 2 L 2m (n). µ µ 2 µ 2 (L 2m (n) 2)µ (L 2m (n) 2)) = 0 µ K 2m (n) = L 2m (n) 2 m > 0 µ 2 K 2m (n)µ K 2m (n) = 0. µ = K 2m (n) + K 2m(n) µ. K 2m (n) < µ µ = K 2m (n)+ µ K 2m (n) = K 2m (n)+ K 2m (n) + K 2m(n) µ K 2m (n) = K 2m (n)+ + µ. L 2m (n) 2 µ = σ 2m (n) = [K 2m (n);, K 2m (n)] = [L 2m (n) 2;, L 2m (n) 2]. m > σ 2m (n) = [L 2m (n) ;, L 2m (n) 2]. n = ϕ m = ϕ 2 = [3 ;, 3 2] = [2; ] = + [; ] = + φ, m > L 2m (n) 2 >

ϕ ϕ 2 = [2; ], ϕ 4 = [6;, 5], ϕ 6 = [7;, 6], ϕ 8 = [46;, 45]. ψ ψ 2 = [5;, 4], ψ 4 = [33;, 32], ψ 6 = [97;, 96], ψ 8 = [53;, 52]. χ χ 2 = [0;, 9], χ 4 = [8;, 6], χ 6 = [297;, 296], χ 8 = [458;, 457]. [a 0 ; a, a 2,...] = [0; a 0, a 2, a 3,...]. ϕ 4 = [0; 6,, 5], ψ 6 = [0; 97,, 96], χ 2 = [0; 0,, 9]. N m n m n α β σ(m) = α + βσ(n). β 0 σ(m) = α m σ(m) N α = 0 β = σ(n) = σ(n) n n n N m n σ(m) = α+βσ(m) α β σ(n) = α + β σ(n) α = α, β = /β n m m n

N m n n r m, n, r σ(m) = α + βσ(n) σ(n) = α + β σ(r) α, β, α, β σ(m) = α + β(α + β σ(r)) = α + β σ(r) α = α + βα, β = ββ. α β m r N [n] = {m N : m n}. [] = {, 4,, 29, 76, 99,...}, [2] = {2, 4, 82, 478, 2786, 6238,...}, [3] = {3, 36, 393, 4287, 46764, 507,...}. [n] σ m (n) m σ(n) σ(n) n n =, 2, 3 n σ(n) n n σ(n) n n σ(n) n = σ(n) σ 2 (n) nσ(n) = σ(n).

n

σ(l 2m+ (n)) = F 2m (n) + F 2m+ (n)σ(n) L 2m+ (n) F 2m (n) F 2m+ (n) n σ(4) = + 2ϕ = ϕ 3 n n n n = 4

x y y = x/σ(n) k = /σ(n) y = σ(n)(x n) k 2 = σ(n) k k 2 = ( /σ(n))σ(n) = σ(n)/(σ 2 (n) + ) /(σ 2 (n) + ) n a(n) = σ(n) σ2 (n) σ 2 (n) +, b(n) = σ2 (n) σ 2 (n) +. a(n)/b(n) = σ(n) n ψ = σ(2) ψ 2 ψ 2 ψ 2 = ((ψ 2) + 2) 2 = 4 + 4(ψ 2) + (ψ 2) 2.

ψ 45 = π/4

ψ / 2 2 2 + = 2 + = ψ. / 2 = (ψ )/2 (ψ+)/2 (ψ + )(ψ )/4 = (ψ 2 )/4 = ψ/2 2ψ

ψ 2ψ a 2a 2 ψ

2 :

x 2 Dy 2 = D D D n = n 2 + 4 x n L 2 m(n) (n 2 + 4)F 2 m(n) = 4( ) m. x 2 D n y 2 = 4( ) m. x m = L m (n), y m = F m (n). x 2 n y 2 = ( ) m,

n 2n x m = L m (2n)/2, y m = F m (2n). n = n 2 + n = m = 2, 4, 6, 8 m =, 3, 5, 7 x 2 2y 2 = ±. x 2 = L 2 (2)/2 = 3, y 2 = F 2 (2) = 2, x 4 = L 4 (2)/2 = 7, y 4 = F 4 (2) = 2, x 6 = L 6 (2)/2 = 99, y 6 = F 6 (2) = 70, x 8 = L 8 (2)/2 = 577, y 8 = F 8 (2) = 408. x 2 2y 2 =. x = L (2)/2 =, y = F (2) =, x 3 = L 3 (2)/2 = 7, y 3 = F 3 (2) = 5, x 5 = L 5 (2)/2 = 4, y 5 = F 5 (2) = 29, x 7 = L 7 (2)/2 = 239, y 7 = F 7 (2) = 69. x 2 2y 2 =. p T p = p(p + ) 2 = q 2 = Q q.

4p 2 + 4p + = 8q 2 +. (2p + ) 2 2(2q) 2 =. x = 2p + y = 2q x 2 2y 2 = p m = L 2m(2) 2 4, q m = F 2m(2). 2 m (p, q ) = (, ), (p 2, q 2 ) = (8, 6), (p 3, q 3 ) = (49, 35), (p 4, q 4 ) = (288, 204). T 8 Q 6 T p Q q

Oxy (±ψ, ±) 2ψ (±/ψ n, ±/ψ n ) n = 0, 2, 4,... (±/ψ n, ±/ψ n ) n =, 3, 5,... (±(ψ )/ψ n, 0), r n = /ψ n

n = 0, 2, 4,... (0, ±(ψ )/ψ n ), r n = /ψ n n =, 3, 5,...

f : [0, ) [0, ) { } x, 0 < x <, f(x) = 0 x = 0. f x {/n, n =, 2, 3,...} f f(x) = f(/n) = 0, x /n 0 f(x) =. x /n+0 f f(0) = 0 f x = 0 (0, ) [0, ) f f m m f 0 (x) = x, f (x) = f(x), f 2 (x) = (f f)(x) = f(f(x)), f 3 (x) = (f f f)(x) = f(f(f(x))),... ξ (0, ) f(ξ) = ξ y = f(x) y = x f ξ n ξ n f ξ f 0 < ξ < ξ ξ = [0; a, a 2, a 3,...].

a, a 2, a 3,... /ξ = a ξ = [a ; a 2, a 3,...] f(ξ) = [0; a 2, a 3,...]. f (a, a 2, a 3,...)

f(ξ) = ξ [0; a 2, a 3,...] = [0; a, a 2, a 3,...], a = a 2 = a 3 =... a n ξ n = [0; n, n, n,...] = [0; n]. ξ n = σ(n) n = σ(n) = n2 + 4 n. 2 σ(n) σ 2m+ (n) = [L 2m+ (n); L 2m+ (n)] = [0; L 2m+(n)].

f m > σ 2m (n) = [L 2m (n) ;, L 2m (n) 2]. σ 2m (n) = [0; L 2m(n),, L 2m (n) 2] f ( ) = [0;, L σ 2m 2m (n) 2]. (n) ( ) f 2 = [0; L σ 2m 2m (n) 2, ], (n) ( ) f 3 = [0;, L σ 2m 2m (n) 2]. (n) L = f ( ) = [0;, L σ 2m 2m (n) 2] (n) f 2 (L) = L, f(l) L, L = f ( ). σ 2m (n) L f m = n = σ() = ϕ f(/ϕ) = /ϕ, f(/ϕ 2 ) = /ϕ, f 2 (/ϕ 2 ) = /ϕ = f(/ϕ 2 ). L = f(/ϕ 2 ) = /ϕ f(l) = L L f

a, b, c b c ξ = [a; b, c] ξ ξ = a + b + c + b + = a + b + c + ξ a = a + c + ξ a bc + bξ ab +. ξ bξ 2 b(2a c)ξ + a 2 b abc c = 0, ξ = 2ab bc + bc(bc + 4). 2b (bc) 2 + 4bc = (bc + 2) 2 4 b c (bc + 2) 2 4 = d 2 d d 2 + 2 2 = (bc + 2) 2 d bc + 2 a =, b = 2, c = 3 ξ = 5 f 2 = [; 2, 3], η = ξ = 5 + 7 = [0;, 2, 3]. 5 3 5 3 5 3 f(η) =, f 2 (η) =, f 3 (η) = = f(η), 2 3 2 ξ f ζ = f(η) f 2 (ζ) = ζ

ξ = 7 4 = [0;, 3, ]. f(ξ) = 7 3, f 2 (ξ) = 4 7 3, f 3 (ξ) = 2 7 4 = ξ. ξ f a, b, c ξ = [0; a, b, c]. ξ ξ = a + b + c + a + = a + b + c + ξ = bc + bξ + abc + abξ + a + c + ξ.

(ab + )ξ 2 + (abc + a b + c)ξ (bc + ) = 0, ξ = b a c abc + (abc + a + b + c) 2 + 4. 2(ab + ) a =, b = 2, c = 3 3ξ 2 + 8ξ 7 = 0 37 4 ξ = = [0;, 2, 3]. 3 f f(ξ) = 37 4, f 2 (ξ) = 3 37 5, f 3 (ξ) = 4 37 4 3 = ξ. ξ f n n 3/4 λ 2 3 4 λ = 0, ξ = 3 + 73 8, η = ξ = 3 + 73. 8 f η 5 + 73 8, 7 + 73 6 7 + 73 8, 5 + 73 4, 7 + 73 2, 5 + 73, 3 + 73 6 8 η f, 7 + 73 2 = η., 5 + 73, 4

f ξ = [0; p, p 2,..., p r, q, q 2,..., q s ], p, p 2,..., p r q, q 2,..., q s ξ ξ f s Πλάτων Άριστοτέλης

Ηρακλῆς Ζεύς Ἀλκμήνη Ηρα Εὐρισθεύς Ολυμπος Αιδης Κέρβερος Αὐγείας Ἀλϕειός, Πηνειός y = a (x/a)

Εὔδοξος ὁ Κνίδιος

Trudy Instituta matematiki i mehaniki UrO RAN

f n (x) = (f(x)) n x f n (x) n f x F m m F m (x) m L m m L m (x) m σ(n) n ϕ = σ() χ = σ(3) ψ = σ(2)