Question

Find all integers a,b,c such that a^2=bc+1 , b^2=ac+1

Answer 1

Given : a² = bc +  1                                                             ---- ( 1 )

And

b² = ca + 1                                                                           ---- ( 2 )

Now we subtract equation 2 from equation 1 and get

a² - b² =  bc - ca

( a + b ) ( a - b ) = c ( b - a )

( a + b ) ( a - c )  = - c ( a - b )

( a + b ) = - c

c = - ( a + b )  , Substitute that value in equation 1 we get

a²  = b [ - ( a + b ) ] + 1

a² = - ab - b² + 1

a² + ab + b² - 1 = 0



So, the integers are ,
a = 0 , b = - 1  and c = 1

a = 0 , b = 1 and c = - 1

a = 1 , b = - 1 and  c = 0

a = 1 , b = 0 and c = - 1

a = - 1 , b = 0 and c = 1

a = - 1 , b = 1 and c = 0