On Coprime Numbers and Age Differences

Question

Hypothetical Situation: Currently, I am $38$ and my wife is $33$, so this year our ages are coprime. Will our ages always be coprime? (If we happen to have the same birthday, so our age difference remains constant).

In general, if any two numbers $n$ and $n+\alpha$ are coprime, will they be coprime for all $n$?

I can see that this is true when $\alpha =1$, obviously. I can also see that $\alpha$ can never be even, because if $n$ and $\alpha$ are even, $n + \alpha$ would also be even, and therefore not coprime.

In this hypothetical situation, $n = 33$ and $\alpha = 5$. I see that my original hypothesis is incorrect, because if $n=10$, then $n+\alpha =15$, and therefore not coprime.

So my conjecture is: $\forall \{n, \alpha \} \in \Bbb{N}$, $n$ and $n+\alpha$ are always coprime iff $n$ and $\alpha$ are coprime. Is this true, or are there more general cases or restrictions?

Answer 1

Your conjecture:

$n$ and $\alpha$ are coprime iff $n$ and $n + \alpha$ are coprime

Taking contrapositives for each conditional statement, we have the equivalent:

$n$ and $\alpha$ share a prime factor iff $n$ and $n + \alpha$ share a prime factor

Let us prove the latter, equivalent formulation.

Forward: If $n$ and $\alpha$ are divisible by the prime $p$, then write $n = Np$ and $\alpha = Ap$.

Still, $n$ is divisible by $p$, and $n + \alpha = Np + Ap = (N+A)p$ is, too.

Reverse: If $n$ and $n + \alpha$ are divisible by the prime $p$, then write $n = Np$ and $n + \alpha = Bp$.

Still, $n$ is divisible by $p$, and $\alpha = (n + \alpha) - n = Bp - Np = (B-N)p$ is, too.

QED

Answer 2

When you are 35 your wife will be 40 and your ages will not be coprime.

Answer 3

Yes, $n$ and $\alpha$ are coprime $\iff n$ and $n+\alpha$ are coprime.

A property of $\gcd$ is that $\gcd(a,b) = \gcd(a+kb, b)$ for any $k\in\mathbb{Z}$. The proof is very easy.

Answer 4

The GCD of two numbers is always a divisor of their difference. Obviously, if $A=da$, and $B=db$, then $A-B=da-db=d(a-b)$. In this case, the difference is $5$, which is a prime, divisible only by $1$ and itself. Not being coprime implies having a GCD greater than $1$. So, for any $n=5k$, your two ages will not be coprime, but rather divisible by $5$.