Let (7 + 4*sqrt(3))^n = p + b?

Question

let (7 + 4*sqrt(3))^n = p + b,

where n and p are integers and b is a proper fraction. prove that

(1 - b)(p + b) = 1.

Answer 1

If this is true, Bhaskar, then RHS is a rational number and LHS is not.

Any power of 7+4rt3 will involve rt3 and hence it is irrational. How can it be a sum of integer and a proper fraction?

If n = -1 the reciprocal of 7+4rt3 = 7- 4rt3 but then still it is not a "proper" fraction. Please check the details and come back.

Answer 2

1) You can present (7+4sqrt3)^n = A + Bsqrt3 and in addition (7-4sqrt3)^n=A - Bsqrt3

Here A and B are the same ( using the binomial theorem)

Important that  their sum = 2A is integer!

2) Next- note that (7-4sqrt3)<1 and so is (7-4sqrt3)^n<1

3) Turn to  (7+4sqrt3)^n + (7+4sqrt3)^n = 2A( integer)

     (7+4sqrt3)^n = 2A – (7+s4qrt3)^n = p + b,  where b<1– fraction

     It is clear that 2A = p +1   and  1 – b =(7-4sqrt3)^n

4) (1-b)(p+b)= (7+sqrt3)^n*( 7+sqrt3)^n= ((7+4sqrt3)(7+4sqrt3))^n= (49-48)^n=1

This task is OK for (3-2sqrt2), (5-2sqrt6) and so on. Hope it helps.