a, b a bc c b a a b a a a a p > p p p 2, 3, 5, 7,, 3, 7, 9, 23, 29, 3, a > p a p [ ] a bp, b p p cq, c, q, < q < p a bp bcq q a <
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1 S(, a) ln( a)
2 a, b a bc c b a a b a a a a p > p p p 2, 3, 5, 7,, 3, 7, 9, 23, 29, 3, a > p a p [ ] a bp, b p p cq, c, q, < q < p a bp bcq q a < q < p p p.2 ( ). [ ] p, p 2,, p n A p p 2 p n + A A p i p, p 2,..., p n A A A p. p A p A p i (p p i p i+ p n ) + A p i p p i, i,..., n 2, 3 A , 3, 5 A , 3, 7 A , 7 A , 5 2, 7 2
3 , 3, 7, 29, 37, 4, 53, 6, 73, 89, 97,... 3, 7,, 9, 23, 3, 43, 47, 59, 67, 7, 79, 83, [ ] p, p 2,, p n 4 A 4p p 2 p n A A q q m A q j A p i q j p i j m q j 4 j + A q q m (4 + ) (4 m + ) 4K + A 4(p p n ) j q j ( ). p a p a p p [ ] j p aj p q j r j aj pq j + r j, r j p. r, r 2,..., r p r j r, j < p a(j ) aj a (pq j + r j ) (pq + r ) p(q j q ). p r, r 2,..., r p, 2,..., p a p ( 2 (p )) (a )(a 2) (a(p )) (pq + r )(pq 2 + r 2 ) (pq p + r p ) pq + r r 2 r p pq + 2 (p ), (a p )( 2 (p )) pq. 2 (p ) p a p p 3
4 .5. p a p b, c ab + pc [ ] b a p 2.4 ab a p p ab pc.6. a 2 p a 2 + p 4 + [ ] a a 2 + p 2 a p pa a 2 + pb p 4 a 2 pb ( ) 2 (a 2 pb) 2 (a 2 ) 2 pb a 4 2 pb a p + pb pa + pb, p(b A) 2 p [ ] p, p 2,, p n 4 + A (2p p 2 p n ) 2 + p A.6 p 4 + A p A p i p p i n n n ab, < a, b < n < a b < n a 2 ab n, a n. n n a, < a n a n a, < a n a a p. p n p n n n p n n p n 4
5 , 3, 5, 7,, 3, 7, a >, n > a n 2.. [ ] x n (x )(x n + x n x + ). f(x) x n + x n x + (x )f(x) xf(x) f(x) x(x n + x n x + ) (x n + x n x + ) x n + x n + + x 2 + x x n x n 2 + x x n a + a + a a n an a. [ ] 2. x a a 0 2. x a a n (a )(a n + a n a + ). a > a, a 2 n n ml, < m, l < n 2. (n m ) x m (x )(x m + x m x + ). 5
6 x 2 l 2 n 2 ml (2 l ) m (2 l ) ( (2 l ) m l + ) a n a 2 n p (p ) q 2 p.4 2 q q q p q p.5 b, c (q )b + pc q 2 (q )b (2 q )((2 q ) b q + ) 2 (q )b 2 +pc 2(2 pc ) + 2(2 p )((2 p ) c p + ) + q 2.2 6, 2, 3, 6 2, 2, 3, 4, 6, > 2. 5, 3, 5, 5. 28, 2, 4, 7, 4, < , 28 n n n n n ( ) + 3( ) ( + 5)
7 (2 2 ), (2 3 ), (2 5 ). q 2 p n 2 p q 2 p (2 p ) q 2 p n 2 p q n, 2,..., 2 p 2, 2 p, q, 2q,..., 2 p 2 q p + q + 2q p 2 q ( p ) + q( p 2 ) 2p 2 + q 2p 2 p + q(2 p ) 2 q + 2 p q q 2 p q n. n ( ) 2.5 ( ). n 2 p (2 p ), q 2 p [ ] n n 2 m,, m m d < < d r < d r m m T T d + + d r n 2 m 2 i d j (0 i, j r) n S 2.2 S d ( ) + + d r ( ) + d r ( ) d (2 + ) + + d r (2 + ) + d r (2 + ) (2 + )(d + + d r ) (2 + )T. n n n n n n S 2n 2 + m 2 + m (2 + )T. 7
8 T 2 + T 2 + c (c ), () m (2 + )c. (2) c > c >, c, m m (2 + )c () T d + + d r + c + m + c + (2 + )c c + T. c (2), () m 2 +, T m. T + m m m m m 2 + m p p x π(x) (x ) 8
9 0 2, 3, 5, 7 π(0) , 3, 5, 7,, 3, 7, 9 π(20) 8 π(30) 30 m 2 3 m 3 5 m 5 6 m 6 0 m 0 5 m 5 30 m 30 m 2 30/2 5, m 3 30/3 0, m 5 30/5 6, m 6 30/6 5 m 0 30/0 3, m 5 30/5 2, m 30 30/ , 3, 5 22 m 2 + m 3 + m 5 m 6 m 0 m 5 + m M M , 3, 5 π(30) M + 3 M ( ) π(0) 4, π(0 2 ) 25, π(0 3 ) 68, π(0 4 ) 229, π(0 5 ) 9592, π(0 6 ) 78498, π(0 7 ) x x π(x) 4 4. H(n) n 9
10 n H(n) a, b (a < b) { D(a, b) (x, y) 0 y } x, a x b S(a, b) y y x D(a, b) O x a x b x : y x y / y /( + ) y /x x x + 2: y x x D(, + ) S(, + ) 2 + < S(, + ) < n n < S(, n) < n. 0
11 S(, n) + n < H(n) < S(, n) +. (3) y 4.2 S(, a) y /x /2 O A A B x x x 2 x 4 3: y x D m(d) 3 m(a) S(, 2) m(b) S(2, 4) 4 A, B y 4 C, D m(c) 4m(A), m(d) 4m(B) 5 F G 2 E m(e) + m(g) m(e) + m(f ) m(d) 4m(B). (4) (4) y y 2 x 4 y ( y 2) x y x x 2 y 4 x ( x 2) C m(e) + m(g) m(c) 4m(A). (5) (4) (5) 4m(A) 4m(B), m(a) m(b)
12 y 4 2 y 4/x C D O x x 2 x 4 4: y 4 x x 2
13 y y 4/x 2 G E F O x 2 x 4 5: y 4 x x m(a) m(b) 2 < m(a) <, < m(b) < 2 2. x 3 A x 3 B < m(a) < , < m(b) < x 2 n x + n, 0,,..., n n + x n < m(a) < x n n + x 0 n + + x n + + x 2 n n + + n n < m(a) < n + n n, 2 x 4 n x n, x n, 0,,..., n 3
14 2 n + 2 x n x n x n < m(b) < 2 n + 2 x 0 n x n n + + n n < m(b) < n + n n, T n n + + n n n + n ( 2n T n + n ) T n + 2n 2n, T n < m(a) < T n + 2n, T n < m(b) < T n + 2n 2n < m(a) m(b) < 2n x n n m(a) m(b) S(, 2) S(2, 4) a, b (a < b) D(a, b) S(a, b) y { (x, y) 0 y } x, a x b, y x D(a, b) O x a x b x 6: y x 4
15 4.. S(a, b) S (, b ) a (0 < a < b). b S(a, ) S (, ) a (0 < a < ). y [ ] b > a > 0, c > 0 y c x c a c b G E F O x a x b x 7: y c x D c (a, b) { (x, y) 0 y c } x, a x b S c (a, b) D c (a, b) y D(a, b) c S c (a, b) cs(a, b). (6) D c (a, b) 7 E, F F m(f ) c (b a) 7 G b m(g) ( c a c b) a c(b a) b c (b a) m(f ) b 5
16 E G { (x, y) 0 x c y, c b y c } a x y ( c D c b, c ) { (x, y) 0 y c a x, c b x c a ( c S c b, c ) a ( c S c (a, b) m(e) + m(f ) m(e) + m(g) S c b, c a ( c b a), c, (6) cs(a, b) cs ( c S(a, b) S b a), c } ). (b > a > 0, c > 0). (7) ( c b S(a, b) S, b ) a [ ] a x b n h b a n x a + h, 0,,..., n x b a n h ( ) b n a x + h, 0,,..., n n h n < S(a, b) < h, x x h x n n ( ) b a h x < S 0 (, b ) n < h a x 0 + h b a n a + ah b a n a + h h x n T n h x n h, x n h (b a) x n 0 h x n n h. x 0 na + (b a) 6
17 n 0 h ( T n + (b a) x na ) T n + nb T n < S(a, b) < T n + (b a)2 nab (b a)2, nab ( (b a)2, T n < S, b ) (b a)2 < T n + nab a nab < S(a, b) S n S(a, b) S (, b ) (b a)2 < a nab (, b ) a 4.2. a > 0 ln(a) S(, a), a >, ln(a) 0, a, S(a, ), 0 < a < (8) 4.3. a, b ( ) ln ln(a), a ( ) b ln ln(b) ln(a), a ln(ab) ln(a) + ln(b). [ ] 0 < a < a > 4. ln(a) + ln(a ) S(a, ) + S(, a ) S(, a ) + S(, a ) 0. a > ln(a) + ln(a ) 0 < a < b 8 S(, b) S(, a) + S(a, b) 4. ( ) ( b ln S, b ) S(a, b) S(, b) S(, a) ln(b) ln(a). (9) a a < a < b ( ) b ln(ab) ln ln(b) ln(a ) ln(b) + ln(a). a 7
18 y y /x O x x a x b 8: S(, a) + S(a, b) S(, b) x 4.4. a n ln(a n ) n ln(a). [ ] n ln(a n ) (n ) ln(a) ln(a n ) ln(a n a) ln(a n ) + ln(a) (n ) ln(a) + ln(a) n ln(a). n ln(a n ) n ln(a) n 0 ln(a 0 ) ln() 0 n < 0 ln(a n ) ln(a n ) ( n) ln(a) n ln(a). x 2 > x > 0 ln(x 2 ) > ln(x ) y ln(x) (x > 0) a > ln(a) S(, a) > n ln(a n ) n ln(a) n ln(a n ) x ln(x) x a > D(, a) a ln(a) S(, a) a ln(a) a (a > ). 0 < a < D(a, ) a S(a, ) a ln(a) S(a, ) ln(a) S(a, ) ( a) a a > 0 ln(a) a ln(x) x < x (x > 0). (0) 8
19 ϵ > 0 n n < ϵ (( ) n ) ln(x) ln x n n ln(x n ) nx n nx ϵ (x ) (0) x + x ln( + x) x (x > ). () 4.5. a > 4.4 ln(a n ) n ln(a) n ln(a n ) y ln(x) ln(e) e e ln(e) > ln( + ) ln(2) e > 2 e ln( a) r < r + r r n rn r rn r r r rn. r < n r n 0 n r n 0 n + r + r r n r 4.6. n0 r n + r + r 2 + r. r
20 : < a < ln( a) S( a, ) a 0 < r < x n ar n (n 0,, 2,...) x 0 a < x < x 2 < < x n < x n+ < < 0 n0 x n+ (x n+ x n ) S( a, ) 0 < ar n < 4.6 x n x n+ (4) 4.6 ar n + (arn ) + (ar n ) 2 + n0 x n (x n+ x n ). (2) (ar n ) 0 ar n+ + (arn+ ) + (ar n+ ) 2 + a r n, (3) 0 a r r n. (4) 0 x n+ x n ( ar n+ ) ( ar n ) a( r)r n 20
21 y y x O x 0 x x 2 x 0: ln( x) (x n+ x n ) x n+ [ a( r) n0 n0 a( r)r n ar n+ ar + r ar + r2 2 ar + r3 3 ar + 4 a( r) [ ( + ar + a 2 r 2 + a 3 r 3 + ) + r( + ar 2 + a 2 r 4 + a 3 r 6 + ) + r 2 ( + ar 3 + a 2 r 6 + a 3 r 9 + ) + r 3 ( + ar 4 + a 2 r 8 + a 3 r 2 + ) + )] a( r) [ ( + ar + a 2 r 2 + a 3 r 3 + ) + (r + ar 3 + a 2 r 5 + a 3 r 7 + ) + (r 2 + ar 5 + a 2 r 8 + a 3 r + ) + (r 3 + ar 7 + a 2 r + a 3 r 5 + ) + ] a( r) [ ( + r + r 2 + r 3 + ) + a(r + r 3 + r 5 + r 7 + ) + a 2 (r 2 + r 5 + r 8 + r + ) + a 3 (r 3 + r 7 + r + r 5 + ) + ] [ a( r) r + ar ] r + a2 r 2 2 r + a3 r 3 3 r + 4 [ a + ar ] + r + a 2 r 2 + r + r + a 3 r r + r 2 + r ] 0 a + r + r + + r.
22 (3) 4.6 ( ) (x n+ x n ) x n n0 a( r) n0 a( r)r n ar n [ a + r ar + r2 ar 2 + r3 ar 3 + a( r) [ ( + a + a 2 + a 3 + ) + r( + ar + a 2 r 2 + a 3 r 3 + ) + r 2 ( + ar 2 + a 2 r 4 + a 3 r 6 + ) + r 3 ( + ar 3 + a 2 r 6 + a 3 r 9 + ) + )] a( r) [ ( + a + a 2 + a 3 + ) + (r + ar 2 + a 2 r 3 + a 3 r 4 + ) + (r 2 + ar 4 + a 2 r 6 + a 3 r 8 + ) + (r 3 + ar 6 + a 2 r 9 + a 3 r 2 + ) + )] a( r) [ ( + r + r 2 + r 3 + ) + a( + r 2 + r 4 + r 6 + ) + a 2 ( + r 3 + r 6 + r 9 + ) + a 3 ( + r 4 + r 8 + r 2 + ) + ] [ a( r) r + a ] r + a2 2 r + a3 3 r + [ 4 a + a ] + r + a 2 + r + r + a r + r 2 + r + 3 a + + r + + r. 0 (2) 0 a + r + r + + r S( a, ) a + + r + + r. (5) 0 < r < 0 ( + )r + r + + r +, r + r + + r + + r + + r. a a + r + r + + r 0 a ] a + + r + + r. (6) (6) T 0 T T 2 T 0 T T 2 22
23 T 0 T T 2 T 0, T 2 T T 2 T 0 0 < a < a + ( r ) 0 T 2 T 0 + r + + r ( r) + + r + + r a + r + + r ( r) a + ( r) a2 0 (r ). a r T 2 T, T 0 T (6) T (5) r S( a, ) T 0 a + + a ln( a) S( a, ) ln( a) S( a, ) a (0 < a < ) ln( x) x x + 2 x2 + 3 x3 + (0 < x < ). ( ) 4.8. ln( + x) ( ) x x 2 x2 + 3 x3 ( < x < ). [ ] x 0 0 < x < ( + x)( x) x 2, 0 < x 2 < 4.7 ln( x) ln( x 2 ) x x 2. x x 2 2, 23
24 ln( x 2 ) ln(( x)( + x)) ln( x) + ln( + x) ln( + x) ln( x 2 ) ln( x) x 2 + x x x 2 2 ( ) x. < x < 0 x x 4.7 x 2 2 x ln( + x) ln( x ) ( x) ( ) x x π(x) ln(x) p, p 2,... (p 2, p 2 3,...) n 2 n m p m n < p m+ P (n) n P (n) p n p p p 2 p m P (n) n P (n) T (n) ( T (n) + + ) ( ) ( ) +. p p 2 p 2 p 2 2 p m p 2 m T (n) r,...,r m 0 p r > p r m 24 n H(n). (7)
25 n p r p r m m ( p j p 2 j ). p j p j T (n) ) ( ( p ) ( ) p 2 p m 4.3 ( ( ln(t (n)) ln ) ) ( ( + ln ) ) ( ( + + ln ) ) p ) ln ( p m j ) ln ( pj. 4.7 ) ln ( pj p 2 ) ln ( p2 ln ( ) + p j p j 2p 2 j ( ) p m + 3p 3 j + p m ) < ln ( pj p j + p j 2p 2 j < + p j 2p 2 j + p j 2p 2 j + p j 2p 2 j + 3p 3 j + 2p 3 j + + 2p 4 j ( ( ) ( ) ) p j p j + p j 2 p j (p j ). p j 25
26 j j m ) < ln ( p p p 2 p m (8), (7) (3) < p + p ) ( ln ( p2 ln ) p m p m + 2 p (p ) + 2 p 2 (p 2 ) p m (p m ) < n + p p 2 p m 2 ( ) n ( + p p 2 p m 2 ) ( ) p p 2 p m 2 n < p p 2 p m 2, P (n) < ln(t (n)) < P (n) + 2. (8) P (n) > ln(t (n)) 2 > ln(h(n)) 2 > ln(s(, n)) 2 ln(ln(n)) n P (n) P (n) > ln(ln(n)) 2 (n 2) n P (n) P (n) 6 n π(n) π(n) 2 n n n! n! n n! p n! p n p n p n! p p n n! p a a a e(p, n) e(p, n) 26
27 6.. x x [x] [x] x < [x] x, y [x] + [y] [x + y] [x] + [y] +. [ ] x [x] + x, y [y] + y, 0 x <, 0 y < [x] + [y] [x] + [y] + x + y x + y < [x] + [y] x + y < [x + y] [x] + [y] x + y < 2 [x + y] [x] + [y] n! n! p n p e(p,n) e(p, n) [ ln(n) ln(p) ] [ ] n p r r [ n r p r ]. [ ] e(p, n) r p r n [ ln(n) ln(p) ] r [ ln(n) ln(p) ] r r#{ a n a p r } ([ ] [ ]) n n r p r p r+ [ ] n r p r [ ln(n) ln(p) ] r [ ln(n) ln(p) ] r [ ] n (r ) p r [ ] n r p r [ ln(n) ln(p) ] r [ n p r [ ln(n) ln(p) ] ( ) n n 2 ( ) n n(n ) (n + ) n!! (n )!! ( ) n 0 ( ) n (0 n) r ]. [ ] n r p r+ 27
28 [ ] [ ] e(p, ) + e(p, n ) + r r p r [ ] n r [ p + n ] r p r p r ([ ] [ ]) n + r p r [ ] n e(p, n). ( ) n n! p p (n )!! n ( ) 2n b n (2n)! n (n!) 2 r b n 6.5. b n 22n 2n (n 2). [ ] n 2 b 2 6 > 24 n 2 3 b n+ p r (2n + 2)! (2n + 2)(2n + )(2n)! (n + )! 2 (n + ) 2 (n!) 2 b n+ 2(2n + ) (n + ) b n > (2n + ) 2 2(n + )(2n ) > 22n+2 2n + 2(2n + ) (n + ) 2 2(n+) 2(n + ). 2 2n 2n 2 2n+2 2n + 4n2 + 4n + 4n 2 + 2n 2 2(2n + ) (n + ) b n 2 2n+2 2n + n + 2 n 6.6. b n (2n) π(2n) p r [ ] b n (2n)!/(n!) b n p e(p,2n) 2e(p,n), p 2n e(p, 2n) 2e(p, n) [ ln(2n) ln(p) ] r [ ] 2n 2 p r [ ln(2n) ln(p) ] r 28 [ ] n p r [ ln(2n) ln(p) ] r ([ ] 2n p [ ]) n 2. p
29 6.2 [ ] 2n p e(p, 2n) 2e(p, n) [ ] n 2 p [ ln(2n) ln(p) ] [ ] ln(2n). ln(p) ln(b n ) (e(p, 2n) 2e(p, n)) ln(p) [ ] ln(2n) ln(p) ln(p) p 2n p 2n ln(2n) ln(p) ln(p) ln(2n) π(2n) ln(2n) p 2n p 2n ln ( (2n) π(2n)). b n (2n) π(2n) n 2 6.5, n 2n < 22n 2n b n (2n) π(2n). 2n ln(2) ln(2n) π(2n) ln(2n), 2n π(2n) ln(2) ln(2n). n 2 2n π(2n) π(2n ) 2n π(2n ) π(2n) ln(2) ln(2n) (2n ) ln(2) ln(2n ) 2n ln(2n ) (2n ) ln(2n). 2n ln(2n ) (2n ) ln(2n) ln(2n ) (2n )(ln(2n) ln(2n )) ( ) 2n ln(2n ) (2n ) ln 2n ( ln(2n ) (2n ) ln + ). 2n 29
30 ( () ln + ) 2n 2n 2n ln(2n ) (2n ) ln(2n) ln(2n ) (2n ) 2n ln(2n ) > (2n ) ln(2n), ln(2n ) ln(3) > 0. 2n 2n ln(2n ) (2n ) ln(2n) > (2n ) π(2n ) ln(2) ln(2n ). n 3 n π(n) ln(2). (9) ln(n) n 2 π(n) ( ) 2n + (2n + )! c n n (n + )!n! c n 6.7. c n < 2 2n (n ). [ ] n c 3 < 2 2 n c n+ (2n + 3)! (n + 2)!(n + )! (2n + 3)(2n + 2)(2n + )! (n + 2)(n + )(n + )!n! 2(2n + 3) n + 2 c n c n+ 2(2n + 3) n + 2 c n < 2(2n + 3) n n 2n + 3 2n n+2 < 2 2n+2 2 2(n+). n + n 6.8. c n p. n+<p 2n+ 30
31 [ ] c n (2n + )! (n + )!n! (2n + )(2n) (n + 2)(n + )! (n + )!n! (2n + )(2n) (n + 2) n! p n + < p 2n + c n p c n n+<p 2n+ p n p c n < 2 2n. x n+<p 2n+ n+<p 2n+ ln(p) < 2n ln(2). θ(x) p x ln(p) n θ(2n + ) θ(n + ) < 2n ln(2). (20) 6.9. θ(n) < 2n ln(2) (n ). [ ] θ() 0 < 2 ln(2), θ(2) ln(2) < 4 ln(2) n, 2 m n 2m (20) θ(2m + ) < θ(m + ) + 2m ln(2) < 2(m + ) ln(2) + 2m ln(2) 2(2m + ) ln(2). n 2m + n 2m + 2 n θ(2m + 2) θ(2m + ) < 2(2m + ) ln(2) < 2(2m + 2) ln(2) n 2m + 2 n 6.0. θ(x) < 2x ln(2) (x ). 3
32 [ ] 6.9 θ(x) θ([x]) < 2[x] ln(2) 2x ln(2). π(x) θ(x) ln(p) ln(p) + p x p x 3/4 ln(2) + ln(x 3/4 ) p x 3/4 x 3/4 <p x x 3/4 <p x ln(p) ln(2)π(x 3/4 ) ln(x)(π(x) π(x3/4 )) 3 4 π(x) ln(x) 3 (ln(x) 43 ) 4 ln(2) π(x 3/4 ). 6.0 x 4 π(x) ln(x) 4 ( 3 θ(x) + ln(x) 4 ) 3 ln(2) π(x 3/4 ) < 8 ln(2) x + (ln(x) 43 ) 3 ln(2) π(x 3/4 ) π(x) ln(x) x 8 ln(2) 3 8 ln(2) 3 < 8 ln(2) 3 x + (ln(x) 43 ) ln(2) x 3/4, + (ln(x) 43 ) ln(2) x /4 + ln(x) x /4 8 ln(2) ln(x/4 ) x /4 < 8 ln(2) e < x 4 2 x 4 π(x) 3.32x ln(x) 3.32x ln(x) 3.32x ln(4) 3.32 ln(2) x 2 π(x) 3.32x ln(x) 3 > π(4) π(x) 32
33 6.. n 2 n ln(2) π(n) 3.32 n ln(n) ln(n) (n 2) ( ). π(x) ( ) (x ). x ln(x) n P (n) 6. x ln() ( 2,..., n) { D(x, x ) (x, y) 0 y } x, x x x S(x, x ) n S(x, x ) S(x 2, x n ) S 3 (, x ) n ln x 2 ( xn /x, x x x 2 S(x, x ) > x (x x ) ) ln(x n ) ln(x 2 ) n 3 x x x < n S(x, x ) ln(x n ) ln(x 2 ) ln ln(n) ln ln(2). (2) 3 x ln() ( ) ( x x ln() ln( ) ln ln ) m > m, x x x > m ln(). 33
34 (2) n 3 c ln() { π(n) < ln ln(n) ln ln(2). (22), 0, n c, P (n) 2 c π() π( ) 6. a 3.32 P (n) n 2 n 2 π(n) n c π() n 2 2 n + 2 an n n ln(n) + π() π( ) n π( ) ( + 2 a n ln(n) + (22) P (n) < n 8 2 ( + ) < a n ln(n) + a ln(). 2 a n ln(n) + 2 n 2 ) π() a ln() n 2 π() c a ( + ) ln() < a ln(n) + n a ln() a ln(n) + 2 n a n 2 ln(2) + < a ln(n) + a + a ln ln(n) a ln ln(2). 2 ln(2) π() + a ( + ) ln() 3 a ln() ln(n) + ln ln(2) 2 ln(2) 3 ln(2) + ln ln(2) 2.29 < 3.3 ln ln(8), ln(2) P (n) < a ln(ln(n)) + 3.3a ln(ln(8)) 4.3a ln(ln(n)) < 4 ln(ln(n)) (n 8) 34
35 6.3. P (n) < 4 ln(ln(n)) (n 8). n 3 35
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熊本県数学問題正解
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