Ta có:
x2+x+2009=y2x2+x+2009=y2
→4x2+4x+8036=4y2→4x2+4x+8036=4y2
→(4x2+4x+1)+8035=(2y)2→(4x2+4x+1)+8035=(2y)2
→(2x+1)2+8035=(2y)2→(2x+1)2+8035=(2y)2
→(2y)2−(2x+1)2=8035→(2y)2−(2x+1)2=8035
→(2y−2x−1)(2y+2x+1)=8035→(2y−2x−1)(2y+2x+1)=8035
→(2y−2x−1,2y+2x+1)→(2y−2x−1,2y+2x+1) là cặp nghiệm của 80358035
Mà 2y−2x−1+2y+2x+1=4y⋮42y−2x−1+2y+2x+1=4y⋮4
→(2y−2x−1,2y+2x+1)∈{(1,8035),(8035,1),(5,1607),(1607,5),(−1,−8035),(−8035,−1),(−5,−1607),(−1607,−5)}→(2y−2x−1,2y+2x+1)∈{(1,8035),(8035,1),(5,1607),(1607,5),(−1,−8035),(−8035,−1),(−5,−1607),(−1607,−5)}

→(2y−2x,2y+2x)∈{(2,8034),(8036,0),(6,1606),(1608,4),(0,−8036),(−8034,−2),(−4,−1608),(−1606,−6)}→(2y−2x,2y+2x)∈{(2,8034),(8036,0),(6,1606),(1608,4),(0,−8036),(−8034,−2),(−4,−1608),(−1606,−6)}

→(y−x,y+x)∈{(1,4017),(4017,0),(3,803),(803,2),(0,−4017),(−4017,−1),(−2,−8004),(−8004,−3)}→(y−x,y+x)∈{(1,4017),(4017,0),(3,803),(803,2),(0,−4017),(−4017,−1),(−2,−8004),(−8004,−3)}

Vì y−x+y+xy−x+y+x chẵn

→(y−x,y+x)∈{(1,4017),(3,803),(−4017,−1),(−2,−8004)}→(y−x,y+x)∈{(1,4017),(3,803),(−4017,−1),(−2,−8004)}

→(2y,2x)∈{(4018,4016),(806,800),(−4018,4016),(−8006,−8002)}→(2y,2x)∈{(4018,4016),(806,800),(−4018,4016),(−8006,−8002)}

→(y,x)∈{(2009,2008),(403,400),(−2009,2008),(−4003,−4001)}
 

Đáp án:(y,x)∈{(2009,2008),(403,400),(−2009,2008),(−4003,−4001)}(y,x)∈{(2009,2008),(403,400),(−2009,2008),(−4003,−4001)}

Giải thích các bước giải:

Ta có:
x2+x+2009=y2x2+x+2009=y2
→4x2+4x+8036=4y2→4x2+4x+8036=4y2
→(4x2+4x+1)+8035=(2y)2→(4x2+4x+1)+8035=(2y)2
→(2x+1)2+8035=(2y)2→(2x+1)2+8035=(2y)2
→(2y)2−(2x+1)2=8035→(2y)2−(2x+1)2=8035
→(2y−2x−1)(2y+2x+1)=8035→(2y−2x−1)(2y+2x+1)=8035
→(2y−2x−1,2y+2x+1)→(2y−2x−1,2y+2x+1) là cặp nghiệm của 80358035
Mà 2y−2x−1+2y+2x+1=4y⋮42y−2x−1+2y+2x+1=4y⋮4
→(2y−2x−1,2y+2x+1)∈{(1,8035),(8035,1),(5,1607),(1607,5),(−1,−8035),(−8035,−1),(−5,−1607),(−1607,−5)}→(2y−2x−1,2y+2x+1)∈{(1,8035),(8035,1),(5,1607),(1607,5),(−1,−8035),(−8035,−1),(−5,−1607),(−1607,−5)}

→(2y−2x,2y+2x)∈{(2,8034),(8036,0),(6,1606),(1608,4),(0,−8036),(−8034,−2),(−4,−1608),(−1606,−6)}→(2y−2x,2y+2x)∈{(2,8034),(8036,0),(6,1606),(1608,4),(0,−8036),(−8034,−2),(−4,−1608),(−1606,−6)}

→(y−x,y+x)∈{(1,4017),(4017,0),(3,803),(803,2),(0,−4017),(−4017,−1),(−2,−8004),(−8004,−3)}→(y−x,y+x)∈{(1,4017),(4017,0),(3,803),(803,2),(0,−4017),(−4017,−1),(−2,−8004),(−8004,−3)}

Vì y−x+y+xy−x+y+x chẵn

→(y−x,y+x)∈{(1,4017),(3,803),(−4017,−1),(−2,−8004)}→(y−x,y+x)∈{(1,4017),(3,803),(−4017,−1),(−2,−8004)}

→(2y,2x)∈{(4018,4016),(806,800),(−4018,4016),(−8006,−8002)}→(2y,2x)∈{(4018,4016),(806,800),(−4018,4016),(−8006,−8002)}

→(y,x)∈{(2009,2008),(403,400),(−2009,2008),(−4003,−4001)}