Special number pairs

Number Theory Level 2

A number pair ( $m, n$ ) is called special, if $m$ and $n$ are positive integers, $m$ and $n$ have the same prime divisors, and $m+1$ and $n+1$ have the same prime divisors too. For example ( $2, 8$ ) is a special number pair.

is it true, that there are infinite many special number pairs?

No, it is false Yes, it is true

1 solution

Sándor Daróczi
Aug 29, 2017

Claim: The number pair $(2^n-2, 2^{2n}-2^{n+1})$ is special for any $n \geq 2$ integer.

Proof: Since $2^n-2=2(2^{n-1}-1)$ and $2^{2n}-2^{n+1}=2^{n+1}(2^{n-1}-1)$ , they must have the same prime divisors, namely $2$ and the prime factors of $2^{n-1}-1$ . Moreover $(2^n-1)^2=2^{2n}-2^{n+1}+1$ , hence $2^n-1$ and $2^{2n}-2^{n+1}+1$ must have the same prime divisors too. Thus by the definition of special number pairs, our claim is true.

According to this there is a unique special number pair for every $n \geq 2$ , hence there are indeed infinitely many of them.

Special number pairs

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