Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/dubna/2013/notes/kconrad-lecture1.pdf Äàòà èçìåíåíèÿ: Sun Jul 21 11:50:07 2013 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 15:01:02 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 5

ABC -



ABC -?
.

21 2013 .






ABC -



ABC - (Masser) (Oesterl´ 1985 . e) (Szpiro).

2012 . (Mochizuki) ABC - 4 .






ABC -





ABC - ABC - . ABC - [...] . .






ABC -



( ) . 7x + 5y = 1, x 2 + y 2 = z 2 , y 2 = x 3 - 2, x 3 - x 2 y - y 3 = 11 : ? , ?






ABC -



( ) . 7x + 5y = 1, x 2 + y 2 = z 2 , y 2 = x 3 - 2, x 3 - x 2 y - y 3 = 11 : ? , ? . x 2 - 7y 2 = 1 Z: (1, 0), (8, 3), (127, 48), (2024, 765), . . .






ABC -



( ) . 7x + 5y = 1, x 2 + y 2 = z 2 , y 2 = x 3 - 2, x 3 - x 2 y - y 3 = 11 : ? , ? . x 2 - 7y 2 = 1 Z: (1, 0), (8, 3), (127, 48), (2024, 765), . . . . x 3 - 7y 3 = 1 : (1, 0), (2, 1). : (1/2, -1/2), (-4/5, -3/5), (-5/4, -3/4), (73/17, 38/17), . . .






ABC -



( ) . 7x + 5y = 1, x 2 + y 2 = z 2 , y 2 = x 3 - 2, x 3 - x 2 y - y 3 = 11 : ? , ? . x 2 - 7y 2 = 1 Z: (1, 0), (8, 3), (127, 48), (2024, 765), . . . . x 3 - 7y 3 = 1 : (1, 0), (2, 1). : (1/2, -1/2), (-4/5, -3/5), (-5/4, -3/4), (73/17, 38/17), . . .






ABC -



y 2 = x 3 + k , k Z - { 0} (1888-1972) . (, 1920) k Z - {0}, y 2 = x 3 + k Z, .. k . (.. , , ).






ABC -



y 2 = x 3 + k , k Z - { 0} (1888-1972) . (, 1920) k Z - {0}, y 2 = x 3 + k Z, .. k . (.. , , ). k . y 2 = x 3 + 24 (-2, ±4), (1, ±5), (10, ±32), (8158, ±736844).






ABC -



y 2 = x 3 + 8 Q: 7 13 , (-2, 0), (1, ±3), (2, ±4), - , ± 4 8 y 433 9765 ,± ,... 121 1331

x

(-2, 0), (1, ±3), (2, ±4), (46, ±312).






ABC -



y 2 = x 3 + 8 Q: 7 13 , (-2, 0), (1, ±3), (2, ±4), - , ± 4 8 y 433 9765 ,± ,... 121 1331

x

(-2, 0), (1, ±3), (2, ±4), (46, ±312). y 2 = x 3 + k Z k = 0, |x | |k |?






ABC -



(, 1971) C > 0, .. y 2 = x 3 + k Z k = 0, |x | C |k |2 . |k |2(1-) (, 1982). |x | C |k |2 C , , C = 25 . 7368442 = 81583 + 24 = C 14,1 2230633472 = 3678063 - 207 = C 8,5 1496516106212 = 281873513 + 1090 = C 23,7






ABC -



(, 1971) C > 0, .. y 2 = x 3 + k Z k = 0, |x | C |k |2 . |k |2(1-) (, 1982). |x | C |k |2 C , , C = 25 . 7368442 = 81583 + 24 = C 14,1 2230633472 = 3678063 - 207 = C 8,5 1496516106212 = 281873513 + 1090 = C 23,7






ABC -



(, 1971) C > 0, .. y 2 = x 3 + k Z k = 0, |x | C |k |2 . |k |2(1-) (, 1982). |x | C |k |2 C , , C = 25 . 7368442 = 81583 + 24 = C 14,1 2230633472 = 3678063 - 207 = C 8,5 1496516106212 = 281873513 + 1090 = C 23,7 4478849284284020423079182 = 58538865167812233 - 1641843 = C 2171,6 , (, 1998).






ABC -



( 1980 .), , |k |2 1 + . > 0 C > 0, .. k Z - {0}, y 2 = x 3 + k Z, |x | C |k |2

(1+)

.






ABC -



( 1980 .), , |k |2 1 + . > 0 C > 0, .. k Z - {0}, y 2 = x 3 + k Z, |x | C |k |2 = 0,1 : |x | C
0 ,1

(1+)

.

|k |2,2 .

7368442 = 81583 + 24 = C0,1 7,5, 2230633472 = 3678063 - 207 = C0,1 2,95, 1496516106212 = 281873513 + 1090 = C0,1 5,8, 4478849284284020423079182 = 58538865167812233 - 1641843 = C0,1 124,0.






ABC -



( 1980 .), , |k |2 1 + . > 0 C > 0, .. k Z - {0}, y 2 = x 3 + k Z, |x | C |k |2 = 0,1 : |x | C
0 ,1

(1+)

.

|k |2,2 .

7368442 = 81583 + 24 = C0,1 7,5, 2230633472 = 3678063 - 207 = C0,1 2,95, 1496516106212 = 281873513 + 1090 = C0,1 5,8, 4478849284284020423079182 = 58538865167812233 - 1641843 = C0,1 124,0. , .






ABC -



o . ( (1630-)) n 3 x n + y n = z n x , y , z . 1994 . ( (1844)) Z+ 8 9. m -y n = 1 Z+, m,n 2, x 32 - 23 = 1. 2002 . x m - y n = 1 , . [...] . . , 1978






ABC -



o . ( (1630-)) n 3 x n + y n = z n x , y , z . 1994 . ( (1844)) Z+ 8 9. m -y n = 1 Z+, m,n 2, x 32 - 23 = 1. 2002 . x m - y n = 1 , . [...] . . , 1978






ABC -



ABC -






ABC -



ABC - . . . n (. radical) rad(n) = p1 p2 · · · pr , ee e n = p11 p22 · · · pr r . . 1) rad(1) = 1 2) rad(252) = rad(22 · 32 · 7) = 2 · 3 · 7 = 42 3) rad(10000) = 10 4) rad(am ) = rad(a)






ABC -



ABC - . . . n (. radical) rad(n) = p1 p2 · · · pr , ee e n = p11 p22 · · · pr r . . 1) rad(1) = 1 2) rad(252) = rad(22 · 32 · 7) = 2 · 3 · 7 = 42 3) rad(10000) = 10 4) rad(am ) = rad(a) . , rad(n) n. , (m, n) .






ABC -



ABC - . . . n (. radical) rad(n) = p1 p2 · · · pr , ee e n = p11 p22 · · · pr r . . 1) rad(1) = 1 2) rad(252) = rad(22 · 32 · 7) = 2 · 3 · 7 = 42 3) rad(10000) = 10 4) rad(am ) = rad(a) . , rad(n) n. , (m, n) .






ABC -



a, b, a + b Z+ , a + b rad(a + b). a + b rad(ab(a + b)), (a,b) = 1. . 3044 (a, b), .. 1 a b 100 (a, b) = 1, a + b rad(ab(a + b)) 7 : (1, 1), (1, 8), (1, 48), (1, 63), (1, 80), (5, 27) (32, 49). a + b rad(ab(a + b)). . a = 1 b = 32 - 1. b 212 , rad(ab(a + b)) = rad(b · 3) a + b > 211 rad(ab(a + b)) 3 b 3 · 3 < 11 (a + b). 11 2 2 rad(ab(a + b)).
10






ABC -



a, b, a + b Z+ , a + b rad(a + b). a + b rad(ab(a + b)), (a,b) = 1. . 3044 (a, b), .. 1 a b 100 (a, b) = 1, a + b rad(ab(a + b)) 7 : (1, 1), (1, 8), (1, 48), (1, 63), (1, 80), (5, 27) (32, 49). a + b rad(ab(a + b)). . a = 1 b = 32 - 1. b 212 , rad(ab(a + b)) = rad(b · 3) a + b > b = 32
n 10

b 3 · 3 < 11 (a + b). 11 2 2

211 rad(ab(a + b)) rad(ab(a + b)). 3 - 1, (a + b)/ rad(ab(a + b)) .






ABC -



a, b, a + b Z+ , a + b rad(a + b). a + b rad(ab(a + b)), (a,b) = 1. . 3044 (a, b), .. 1 a b 100 (a, b) = 1, a + b rad(ab(a + b)) 7 : (1, 1), (1, 8), (1, 48), (1, 63), (1, 80), (5, 27) (32, 49). a + b rad(ab(a + b)). . a = 1 b = 32 - 1. b 212 , rad(ab(a + b)) = rad(b · 3) a + b > b = 32
n 10

b 3 · 3 < 11 (a + b). 11 2 2

211 rad(ab(a + b)) rad(ab(a + b)). 3 - 1, (a + b)/ rad(ab(a + b)) .






ABC -



ABC - . ABC - . (a, b, c ), .. a + b = c (a, b, c ) = 1 ( (a, b) = 1). : c > rad(abc ) . ? ABC -, .. c > rad(abc ), c < rad(abc )2 , , 3, c < rad(abc )1,6 , , 13, c < rad(abc )1,5 .






ABC -



ABC - . ABC - . (a, b, c ), .. a + b = c (a, b, c ) = 1 ( (a, b) = 1). : c > rad(abc ) . ? ABC -, .. c > rad(abc ), c < rad(abc )2 , , 3, c < rad(abc )1,6 , , 13, c < rad(abc )1,5 . (, , 1985) > 0 ABC - (a, b, c ), , c < rad(abc )1+ .






ABC -



ABC - . ABC - . (a, b, c ), .. a + b = c (a, b, c ) = 1 ( (a, b) = 1). : c > rad(abc ) . ? ABC -, .. c > rad(abc ), c < rad(abc )2 , , 3, c < rad(abc )1,6 , , 13, c < rad(abc )1,5 . (, , 1985) > 0 ABC - (a, b, c ), , c < rad(abc )1+ . : . a + b = c , (a, b) = 1. : 26 +34
. . . . . . 1 · 291 , 24 · 35 +311 · 76 =5

· 113 = 1731 · 24591 · 114111 .






ABC -



ABC - . ABC - . (a, b, c ), .. a + b = c (a, b, c ) = 1 ( (a, b) = 1). : c > rad(abc ) . ? ABC -, .. c > rad(abc ), c < rad(abc )2 , , 3, c < rad(abc )1,6 , , 13, c < rad(abc )1,5 . (, , 1985) > 0 ABC - (a, b, c ), , c < rad(abc )1+ . : . a + b = c , (a, b) = 1. : 26 +34
. . . . . . 1 · 291 , 24 · 35 +311 · 76 =5

· 113 = 1731 · 24591 · 114111 .






ABC -



ABC - . ABC -, a + b = c (a, b, c ) = 1, a, b, c > 0 a, b, c = 0 . n < 0, rad(n) = rad(|n|). ( a, b, c ) > 0 ABC - (a, b, c ), , max(|a|, |b|, |c |) < rad(abc )1+ . ( ) > 0 > 0, .. ABC - (a, b, c ), max(|a|, |b|, |c |) < rad(abc )1+ .






ABC -



ABC - . ABC -, a + b = c (a, b, c ) = 1, a, b, c > 0 a, b, c = 0 . n < 0, rad(n) = rad(|n|). ( a, b, c ) > 0 ABC - (a, b, c ), , max(|a|, |b|, |c |) < rad(abc )1+ . ( ) > 0 > 0, .. ABC - (a, b, c ), max(|a|, |b|, |c |) < rad(abc )1+ .






ABC -



ABC - f (t ) ( C, ), rad(f ) . . f (t ) = t (1 - t )3 (1 + t )2 , rad(f ) = t (t - 1)(t + 1). ( (1983), (1981)) f (t ) + g (t ) = h(t ), f , g , h , , , max(deg f , deg g , deg h) deg(rad(fgh)) - 1. ABC -. (ln |n| deg f ): ABC -, , max(|a|, |b|, |c |) < rad(abc )1
+

ln(max(|a|, |b|, |c |)) < (1 + ) ln (rad(abc )) max(ln |a|, ln |b|, ln |c |) < (1 + ) ln(rad(abc )). = 0 -1.






ABC -



ABC - f (t ) ( C, ), rad(f ) . . f (t ) = t (1 - t )3 (1 + t )2 , rad(f ) = t (t - 1)(t + 1). ( (1983), (1981)) f (t ) + g (t ) = h(t ), f , g , h , , , max(deg f , deg g , deg h) deg(rad(fgh)) - 1. ABC -. (ln |n| deg f ): ABC -, , max(|a|, |b|, |c |) < rad(abc )1
+

ln(max(|a|, |b|, |c |)) < (1 + ) ln (rad(abc )) max(ln |a|, ln |b|, ln |c |) < (1 + ) ln(rad(abc )). = 0 -1.






ABC -



ABC -






ABC -



ABC -

> 0 ABC - (a, b, c ), , max(|a|, |b|, |c |) < rad(abc )1+ . -: ( ), (, < 1/5), . .






ABC -



ABC -

> 0 ABC - (a, b, c ), , max(|a|, |b|, |c |) < rad(abc )1+ . -: ( ), (, < 1/5), . .






ABC -



, ABC - : max(|a|, |b|, |c |) < rad(abc )1+ ABC -, . x n + y n = z n , n 3 x , y , z Z+ , , n . , (x , y ) = 1, (x n , y n , z n ) ABC -.






ABC -



, ABC - : max(|a|, |b|, |c |) < rad(abc )1+ ABC -, . x n + y n = z n , n 3 x , y , z Z+ , , n . , (x , y ) = 1, (x n , y n , z n ) ABC -. (x n , y n , z n ), , z
n

< rad(x n y n z n ) = rad(xyz )1 (xyz )
1+ +

1+






ABC -



, ABC - : max(|a|, |b|, |c |) < rad(abc )1+ ABC -, . x n + y n = z n , n 3 x , y , z Z+ , , n . , (x , y ) = 1, (x n , y n , z n ) ABC -. (x n , y n , z n ), , z
n

< rad(x n y n z n ) = rad(xyz )1 (xyz ) 1+ 3(1+) +

1+

n < 3(1 + ).






ABC -



, ABC - : max(|a|, |b|, |c |) < rad(abc )1+ ABC -, . x n + y n = z n , n 3 x , y , z Z+ , , n . , (x , y ) = 1, (x n , y n , z n ) ABC -. (x n , y n , z n ), , z
n

< rad(x n y n z n ) = rad(xyz )1 (xyz ) 1+ 3(1+) +

1+

n < 3(1 + ). z n z > 1 n : max(|a|, |b|, |c |) < rad(abc )1+ , n, .






ABC -



, f (t )n + g (t )n = h(t )n , f , g , h . , n < 3. , f , g , h . , deg f n , deg g n , deg hn deg(rad(f n g n hn )) - 1 = deg(rad(fgh)) - 1.






ABC -



, f (t )n + g (t )n = h(t )n , f , g , h . , n < 3. , f , g , h . , deg f n , deg g n , deg hn deg(rad(f n g n hn )) - 1 = deg(rad(fgh)) - 1. deg(rad(fgh)) deg(fgh), n deg f , n deg g , n deg h deg(fgh) - 1.






ABC -



, f (t )n + g (t )n = h(t )n , f , g , h . , n < 3. , f , g , h . , deg f n , deg g n , deg hn deg(rad(f n g n hn )) - 1 = deg(rad(fgh)) - 1. deg(rad(fgh)) deg(fgh), n deg f , n deg g , n deg h deg(fgh) - 1. : n deg(fgh) 3(deg(fgh) - 1) < 3 deg(fgh). , n < 3. . XIX . .






ABC -



, f (t )n + g (t )n = h(t )n , f , g , h . , n < 3. , f , g , h . , deg f n , deg g n , deg hn deg(rad(f n g n hn )) - 1 = deg(rad(fgh)) - 1. deg(rad(fgh)) deg(fgh), n deg f , n deg g , n deg h deg(fgh) - 1. : n deg(fgh) 3(deg(fgh) - 1) < 3 deg(fgh). , n < 3. . XIX . .






ABC -



ABC - ABC - < 1/5 , x m - y n = 1 Z+ , m, n 2. . : f (t )m - g (t )n = 1 m, n 2, f g .






ABC -



ABC - ABC - < 1/5 , x m - y n = 1 Z+ , m, n 2. . : f (t )m - g (t )n = 1 m, n 2, f g . -: , f g . f , g , (f (t )m , -g (t )n , 1), deg f m , deg g n deg(rad(f m g n )) - 1 = deg(rad(fg )) - 1, m deg f , n deg g < deg(fg ) = deg f + deg g .






ABC -



ABC - ABC - < 1/5 , x m - y n = 1 Z+ , m, n 2. . : f (t )m - g (t )n = 1 m, n 2, f g . -: , f g . f , g , (f (t )m , -g (t )n , 1), deg f m , deg g n deg(rad(f m g n )) - 1 = deg(rad(fg )) - 1, m deg f , n deg g < deg(fg ) = deg f + deg g . deg f < deg g deg f , deg g < , m-1 n-1






ABC -



ABC - ABC - < 1/5 , x m - y n = 1 Z+ , m, n 2. . : f (t )m - g (t )n = 1 m, n 2, f g . -: , f g . f , g , (f (t )m , -g (t )n , 1), deg f m , deg g n deg(rad(f m g n )) - 1 = deg(rad(fg )) - 1, m deg f , n deg g < deg(fg ) = deg f + deg g . deg f < deg f < deg g deg f , deg g < , m-1 n-1

deg f deg f = deg f < deg f . (m - 1)(n - 1) , f g .






ABC -



ABC - ABC - < 1/5 , x m - y n = 1 Z+ , m, n 2. . : f (t )m - g (t )n = 1 m, n 2, f g . -: , f g . f , g , (f (t )m , -g (t )n , 1), deg f m , deg g n deg(rad(f m g n )) - 1 = deg(rad(fg )) - 1, m deg f , n deg g < deg(fg ) = deg f + deg g . deg f < deg f < deg g deg f , deg g < , m-1 n-1

deg f deg f = deg f < deg f . (m - 1)(n - 1) , f g .






ABC -



(, 1965) g (t )2 = f (t )3 + k (t ), k (t ) , f (t ) g (t ) , deg f (t ) 2(deg k (t ) - 1). : y 2 = x 3 + k , k = 0 Z |x | C |k |2(1+) . -. , , f (t ) g (t ) . g (t )2 = f (t )3 + k (t ) -






ABC -



(, 1965) g (t )2 = f (t )3 + k (t ), k (t ) , f (t ) g (t ) , deg f (t ) 2(deg k (t ) - 1). : y 2 = x 3 + k , k = 0 Z |x | C |k |2(1+) . -. , , f (t ) g (t ) . g (t )2 = f (t )3 + k (t ) -, deg f (t )3 , deg g (t )
2

deg rad(f (t )3 g (t )2 k (t )) - 1 deg(f (t )g (t )k (t )) - 1.






ABC -



(, 1965) g (t )2 = f (t )3 + k (t ), k (t ) , f (t ) g (t ) , deg f (t ) 2(deg k (t ) - 1). : y 2 = x 3 + k , k = 0 Z |x | C |k |2(1+) . -. , , f (t ) g (t ) . g (t )2 = f (t )3 + k (t ) -, deg f (t )3 , deg g (t )
2

deg rad(f (t )3 g (t )2 k (t )) - 1 deg(f (t )g (t )k (t )) - 1.

3 deg f , 2 deg g deg f + deg g + deg k - 1, 2 deg f deg g + (deg k - 1), deg g deg f + (deg k - 1).






ABC -



(, 1965) g (t )2 = f (t )3 + k (t ), k (t ) , f (t ) g (t ) , deg f (t ) 2(deg k (t ) - 1). : y 2 = x 3 + k , k = 0 Z |x | C |k |2(1+) . -. , , f (t ) g (t ) . g (t )2 = f (t )3 + k (t ) -, deg f (t )3 , deg g (t )
2

deg rad(f (t )3 g (t )2 k (t )) - 1 deg(f (t )g (t )k (t )) - 1.

3 deg f , 2 deg g deg f + deg g + deg k - 1, 2 deg f deg g + (deg k - 1), deg g deg f + (deg k - 1). : 2 deg f deg f + 2(deg k - 1), deg f 2(deg k - 1).






ABC -



(, 1965) g (t )2 = f (t )3 + k (t ), k (t ) , f (t ) g (t ) , deg f (t ) 2(deg k (t ) - 1). : y 2 = x 3 + k , k = 0 Z |x | C |k |2(1+) . -. , , f (t ) g (t ) . g (t )2 = f (t )3 + k (t ) -, deg f (t )3 , deg g (t )
2

deg rad(f (t )3 g (t )2 k (t )) - 1 deg(f (t )g (t )k (t )) - 1.

3 deg f , 2 deg g deg f + deg g + deg k - 1, 2 deg f deg g + (deg k - 1), deg g deg f + (deg k - 1). : 2 deg f deg f + 2(deg k - 1), deg f 2(deg k - 1).






ABC -



ABC - ( ABC ) ABC , > 0 C > 0, .. y 2 = x 3 + k Z k = 0, |x | C |k |2(1+) . ( ABC . )

ABC , > 0 C > 0, .. y 2 = x 3 + k Z, k = 0, (x , y ) = 1, |x | C rad(k )2(1+) . , , (x , y )=1, rad(k ) , |k |. . ABC : . , , !






ABC -



ABC - ( ABC ) ABC , > 0 C > 0, .. y 2 = x 3 + k Z k = 0, |x | C |k |2(1+) . ( ABC . )

ABC , > 0 C > 0, .. y 2 = x 3 + k Z, k = 0, (x , y ) = 1, |x | C rad(k )2(1+) . , , (x , y )=1, rad(k ) , |k |. . ABC : . , , !






ABC -



R, a/b Q ( ) - . = 5 2- a 1 < 2. b b

5 2 1,1486.
2

1148 1 1000 1000 309 1 5 2- < 269 2692

(0,0006

0,000001),

(0,0000005 < 0,0000138). 5 2- a 1 < 2, b b

, b > 1:

7 8 15 23 31 54 85 139 224 309 , , , , , , , , , ,.... 6 7 13 20 27 47 74 121 195 269






ABC -



R, a/b Q ( ) - . = 5 2- a 1 < 2. b b

5 2 1,1486.
2

1148 1 1000 1000 309 1 5 2- < 269 2692

(0,0006

0,000001),

(0,0000005 < 0,0000138). 5 2- a 1 < 2, b b

, b > 1:

7 8 15 23 31 54 85 139 224 309 , , , , , , , , , ,.... 6 7 13 20 27 47 74 121 195 269






ABC -



1 ai ai 5 < 2 () , .. 2- bi bi bi 1 ai 5 2- = 2+i , i > 0. b1 < b2 < b3 < · · · . bi bi i 20 (i = 1, 2, . . . ).






ABC -



1 ai ai 5 < 2 () , .. 2- bi bi bi 1 ai 5 2- = 2+i , i > 0. b1 < b2 < b3 < · · · . bi bi i 20 (i = 1, 2, . . . ).

i , i 0. , 2 + i 2.






ABC -



1 ai ai 5 < 2 () , .. 2- bi bi bi 1 ai 5 2- = 2+i , i > 0. b1 < b2 < b3 < · · · . bi bi i 20 (i = 1, 2, . . . ).

i , i 0. , 2 + i 2. (, 1955) , > 0 a a 1 , , - 2+ . b b |b | .






ABC -



ABC - ( ABC -) > 0 ABC - (a, b, c ), , max(|a|, |b|, |c |) < rad(abc )1+ . ( rad(abc )) > 0 ABC - (a, b, c ), , rad(abc ) > max(|a|, |b|, |c |)1- . ( 1 .) 2 ABC , > 0, rad(a5 - 2b5 ) > max(|a|, |b|)3- a b, ; = 5 2.






ABC -



ABC - ( ABC -) > 0 ABC - (a, b, c ), , max(|a|, |b|, |c |) < rad(abc )1+ . ( rad(abc )) > 0 ABC - (a, b, c ), , rad(abc ) > max(|a|, |b|, |c |)1- . ( 1 .) 2 ABC , > 0, rad(a5 - 2b5 ) > max(|a|, |b|)3- a b, ; = 5 2.

, , ABC - , ABC .






ABC -



ABC - ( ABC -) > 0 ABC - (a, b, c ), , max(|a|, |b|, |c |) < rad(abc )1+ . ( rad(abc )) > 0 ABC - (a, b, c ), , rad(abc ) > max(|a|, |b|, |c |)1- . ( 1 .) 2 ABC , > 0, rad(a5 - 2b5 ) > max(|a|, |b|)3- a b, ; = 5 2.

, , ABC - , ABC .






ABC -



FAQ ABC - a + b = c? max(|a|, |b|, |c |) < rad(abc )1+ ( )?

1

2

1) a + b = c y 2 = x (x - a)(x + b). , ABC -, .






ABC -



FAQ ABC - a + b = c? max(|a|, |b|, |c |) < rad(abc )1+ ( )?

1

2

1) a + b = c y 2 = x (x - a)(x + b). , ABC -, . 2) . , ABC -, ( 1.10 4 ); , . , ABC -.






ABC -



FAQ ABC - a + b = c? max(|a|, |b|, |c |) < rad(abc )1+ ( )?

1

2

1) a + b = c y 2 = x (x - a)(x + b). , ABC -, . 2) . , ABC -, ( 1.10 4 ); , . , ABC -.






ABC -



?






ABC -



. , , http://www.sci- lib.org/books_1/L/leng.pdf. E. Bombieri W. Gubler, Heights in Diophantine Geometry. V. Dimitrov, http://mathoverflow.net/questions/106560/ philosophy- behind- mochizukis- work- on- the- abc- conjecture A. Granville T. Tucker, "It's as easy as abc ", Notices AMS, 2002. S. Lang, "Old and New Conjectured Diophantine Inequalities", Bull. Amer. Math. Society, 1990. A. Nitaj, "La conjecture abc", Enseign. Math., 1996. A. Nitaj, http://www.math.unicaen.fr/~nitaj/abc.html ( ABC -). M. Waldschmidt, "Perfect Powers: Pillai's works and their developments", http://www.math.jussieu.fr/~miw/articles/ pdf/PerfectPowers.pdf.