T = Z t T

t T

t T t T t T : Z N

(s i ) n i=1 n n S S = {(s i ) n i=1 N n : s j + j s k + k ( n), n N}. 1 j k n (s 1, s 2,..., s n ) s 1 s 2... s n m = s 1 s 2... s n m s i m i = 1,..., n S m

S m = {(s i ) n i=1 S : s i m, n N}. 1 i n = s 1 s 2... s n n n n S n = s 1 s 2... s n a a S as = {(s i ) n i=1 S : n s i /n = a, n N}. i=1 S m n, a S m, a S n, a S m n

321 3 + 1 2 + 2 ( 3) = a 1 a 2... a n π : {1, 2,..., n} {1, 2,..., n} a π(1) a π(2)... a π(n) S = a N as = S n, n N S m = ṁ a=0 as m.

a Sm S m a Sm n S m n a S S a S n S n a, n m n a m S, S n, S m, a S a S m S m n, a S n a S m n a, aa, aaa,... a S m a S m = ℵ 0 S = S m = a S = a S m = ℵ 0 000 }{{... 0 }, 111 }{{... 1 }, 222 }{{... 2 },... n n n S n S n = ℵ 0 a S m n S m n mn < ℵ 0 a S n an 2 < ℵ 0 n a a S n

a n a S n (a + 1) n a n = s 1 s 2... s n n a z = s 1 s 2... s n = (s 1 + z)(s 2 + z)... (s n + z) a + z n (s 1 +n)s 2... s n, s 1 (s 2 +n)... s n,..., s 1 s 2... (s n + n) a + 1 (s 1 n)s 2... s n, s 1 (s 2 n)... s n,..., s 1 s 2... (s n n) a 1 s i n = a 1 a 2... a n i j i < j n j i a i S i,j () a j + j i, : k = i, (S i,j ()) k = a i + i j, : k = j, a k, :

S 1 i,j S 1 i,j = S j,i. O = a 1 a 2... a n 1 a n O() = a n a 1 a 2... a n 1. O 1 O 1 () = a 2 a 3 a 4... a n a 1. O(O 1 ) = O 1 (O) = O n = O n = = (b 1 b 2... b n ) m a 1 = m a 2 < m a 1 a 2 = 1

= (b 1 b 2... b n ) = (aa... a) a n n

µ n p i n = p e 1 1 p e 2 2 p er r, µ : Z + { 1, 0, 1} { ( 1) r, e i µ(n) = 0, n > 0 n M n (x) M n (x) = µ(n/d)x d. d n n a 1 n (M n(a + 1) M n (a)) f : Z Z Z x Z f(x) x df : Z Z df(x) = f(x) x x Z

f n i j ( n) = df(i) = df(j). i,j Z f Z a, b Z a b b = f n (a). n Z Z (f)

f df (f) = I x I df(x), I I = {a, a + 1,..., b} Z df M = {df(x) : x Z} I = [a, b] I > M I O I + 1 M x I O df(x) I 1 + M. I (f)( I + 1 M) x I df(x) (f)( I 1 + M). I I I I

a m a m a M m m a ϕ : a S m a M m = s 1 s 2... s n a S m ϕ() = (ϕ()) 1 (ϕ()) 2... (ϕ()) m a M m (ϕ()) i = 1 j {1,...,n} k N i = s j (n j) (kn). a S m m m

ϕ(51) = 11010 ϕ(7531) = 1111000 = s 1 s 2... s n, O 1 (), O 2 (),..., O n+1 () ϕ(), ϕ(o 1 ()), ϕ(o 2 ()),..., ϕ(o n+1 ()). = s 1 s 2... s n ϕ() ϕ(o 1 ()) ϕ(o 2 ()) ϕ(o n+1 ()) ϕ() s 1 s 2... s n a m a < m a M m = ( ) m a

2 S 4 a m a < m a G m = ( a M m, E) a S m = M 1 M 2... M m M 1 = 0 0 O 1 () = M 2 M 3... M m 0 a m a < m a Sm

3 S 5 a Gm m S m m S m = m a=0 as m a S m a m a S m ϕ a M m ( ) m a ϕ a S m ϕ

a G m ϕ a Gm ϕ ϕ a m a < m = s 1 s 2... s n a S m = s 1s 2... s i = s s 1 s 2... s n = s 1s 2... s is 1s 1s 2... s j a S m = 2... s j a m a < m a S m

a S m a m a < m a G m ( m a ) a m a < m a G m 0 m 0 1

φ n φ : N N ) ) ) φ(n) = n (1 (1 1p1 1p2... (1 1pk p 1, p 2,... p k n φ(n) n n a m a < m a S m Ω(a, m) = ( ) m 1 a m d (a,m) ( ) m/d φ(d). a/d ( m ) a

m a a a > 2 a S 2a a S 2a a G 2a 1010... 10 0101... 01 1010... 10 1010... 10 100101... 01 10M 3 M 4... M 2a 100101... 01 1010... 10 M 1 M 2... M 2a 2 10 2a 1 0101... 010110 100101... 01 0101... 010110 a > 2

a m a < m a S m 1 = M 1,1 M 1,2... M 1,m, 2 = M 2,1 M 2,2... M 2,m, Ω(a,m) = M Ω(a,m),1 M Ω(a,m),2... M Ω(a,m),m. 1 = M 1,m M 1,m 1... M 1,1, 2 = M 2,m M 2,m 1... M 2,1, Ω(a,m) = M Ω(a,m),m M Ω(a,m),m 1... M Ω(a,m),1, = s 1 s 2... s Ω(a,m) m a m a < m = s 1 s 2... s n 1 s n a S m = (m s n )(m s n 1 )... (m s 2 )(m s 1 ) m a S m m 2a

m = 2a a = 3

a m

= S 1 S 2... S n = [s 1 1s 2 1... s d 1 1 ][s 1 2s 2 2... s d 2 2 ]... [s 1 n, s 2 n... s d n n ] S i = {s k j : s k j + j ( n) = i, j {1,..., n}, k {1,..., d j }}. i {1,...,n} M

a a = 1 n n d i s j i. i=1 j=1 a M m m m s j i M n a M

M m n, a M m, a M n, a M m n a m a < m a M m ϕ() m a M m m N = S 1 S 2... S n = S 1S 2... S n M n = (S 1 S 1)(S 2 S 2)... (S n S n). a M n a M n a+a M n = S 1 S 2... S n = S 1S 2... S l c = (n, l) = ( }{{... } ) ( } {{... } c n c l ).

a M n a M n a+a M (n,n ) [54][2][4] = 504 420 [54][22][2] = 522 420 [333] = 3 3 3 [43][2][3] = 423 300 = 420 303

U = {(s i ) n i=1 Z n : s j + j s k + k ( n), n N}. 1 j k n

E = mc 2

= a 1 a 2... a n = b 1 b 2... b m = a 1 a 2... a n b 1 b 2... b m. a Sm a Mm a Sm = { a S m : ϕ() = }, a Mm = { a M m : ϕ() = }. a m a < m a S m a M m a Sm a Mm λ ( a Sm, λ, ) ( a Mm, λ, )

M e (M, e, )