Question

show that there is no positive integer p for which root p+1 +root p+3 is rational

Answer 1

Answer:

P/Q=  √(p+1) and √(p+3)                        ------ 1

Q/P=1/ √(p+1)+√(p+3) = √(p+1)√(p+3) 

2q/p=            √(p+1)+√(p+3)                           ------- 2

adding 1 and 2 

\sqrt{p+1} [/tex] = p/q + 2q/p = p +  /2pq --3

subtracting 1 from 2

 

=   ------- 4

from 3 and 4

and  are rational numbers as  and  

are rational for integer p and q .

here n+1 and n-1 are perfect square of positive integer.

now,(n+1)-(n-1)=2 which is not possible since any perfect square differ by

 

at least 3.

thus there is no positive integer p which √(p+1) and √(p+3) is rational.