Question

Do there exist real numbers `a,b` such that (1) `a+b` is rational and `a^n+b^n` is irrational for each natural `n ge 2` (2) `a+b` is irrational and `a^n+b^n` is rational for each natural `n ge 2`?

Answer 1

(1) Yes, e.g. `a=pi`, `b=1-pi`

(2) No, see explanation.

Explanation:

(1) Let `a=pi`, `b=1-pi`

Then:

`a+b = 1` is rational

When `n >= 2`

`a^n+b^n = pi^n+(1-pi)^n`

is a non-trivial polynomial in `pi`, hence irrational, since `pi` is transcendental.

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(2) Suppose `a^n+b^n` is rational for all `n >= 2`

If `a=b=0` then `a+b` is rational.

If `a=0` and `b!=0` or `a!=0` and `b=0` then:

`(a^3+b^3)/(a^2+b^2) = a+b" "` is rational

Otherwise suppose `a, b != 0`

Then:

`(a^2+b^2)^2-(a^4+b^4) = 2a^2b^2`

The left hand side is rational, so the right hand side must be.

Hence `a^2b^2` is rational.

Then:

`(a^2+b^2)(a^3+b^3)-(a^5+b^5) = a^2b^2(a+b)`

The left hand side of this equation is rational and the multiplier `a^2b^2` is also rational (and non-zero). So `a+b` is rational.

Answer 2

(1) Let `a=sqrt(2)` and `b=1-sqrt(2)`

Explanation:

Here's an alternative answer for (1) using simple irrational numbers:

Let `a = sqrt(2)`, `b=1-sqrt(2)`

Then:

`a+b = 1" "` is rational

Note that:

`(p+qsqrt(2))(1-sqrt(2)) = (p-2q)+(q-p)sqrt(2)`

So consider the recursively defined sequences:

`p_0 = 1`

`q_0 = 0`

`p_(n+1) = p_n-2q_n`

`q_(n+1) = q_n-p_n`

Then:

• `p_n + q_nsqrt(2) = (1-sqrt(2))^n` for `n >= 0`

• The sequences begin:

`1, 1, 3, 7, 25, 57, 139,...`

`0, -1, -2, -9, -16, -41, -98,...`

• `p_n > 0` for all `n >= 0` and is always odd.

• `q_n < 0` for all `n >= 1` and alternates between odd and even values.

We can also define sequences for the `a^n` term, by:

`r_0 = 1`

`s_0 = 0`

`r_(n+1) = 2s_n`

`s_(n+1) = r_n`

Then:

• `r_n + s_nsqrt(2) = sqrt(2)^n` for `n >= 0`

• These sequences begin:

`1, 0, 2, 0, 4, 0, 8,...`

`0, 1, 0, 2, 0, 4, 0,...`

Note that when `n >= 2`, both `r_n` and `s_n` are even.

When `n` is even then `s_n = 0`, so `q_n+s_n = q_n < 0` for `n >= 2`.

When `n` is odd then `q_n` is odd so `q_n+s_n` is odd and non-zero.

Hence in all cases, when `n >= 2` we find:

`sqrt(2)^n + (1-sqrt(2))^n = (p_n+r_n) + (q_n+s_n)sqrt(2)`

with `q_n+s_n != 0` and hence irrational.