$2n+1$ and $n^2+1$ are always coprime or their gcd is $5$

Question

Using a spreadsheet, it can be inferred that when $n≡2[5]$, then $\gcd(n^2+1,2n+1)=5$, else $\gcd(n^2+1,2n+1)=1$.

Indeed, when $n≡2[5]$, $n^2+1$ and $2n+1$ can easily be shown to be multiples of $5$, so their gcd is at least $5$. But then, I can't see how to complete the proof.

Answer 1

Note that $4(n^2+1)-(2n-1)(2n+1)=5$, Since the greatest common divisor divides any linear combination $n^2+1,2n+1$ , therefore $gcd(n^2+1,2n+1)|5$

Answer 2

As $(a,b)=(a,a-nb),$

$(2(n^2+1), 2n+1)=(2(n^2+1)-n(2n+1), 2n+1)=(2-n,2n+1)=(2-n,2n+1+2(2n-1))=(2-n,5)$

So, $(2(n^2+1), 2n+1)=5$ if $5\mid(2-n)$ i.e., if $n\equiv2\pmod 5$

$\implies (n^2+1, 2n+1)=5$ as $(2,2n+1)=1$ is as $(2n+1)$ odd.

If $n\not\equiv2\pmod 5,(2(n^2+1), 2n+1)=1\implies (n^2+1, 2n+1)=1$

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Alternatively,

We know, $(a,b)\mid (ax+by)$ where $a,b,x,y$ are integers.

So, $(n^2+1, 2n+1)\mid \{n(2n+1)-2(n^2+1)\}\implies (n^2+1, 2n+1)\mid(n-2)$

Again, $(2n+1,n-2)\mid\{2n+1-2(n-2)\} \implies (2n+1,n-2)\mid 5$

If $n\equiv2\pmod 5,2n+1\equiv0 \pmod 5$ and $n^2+1\equiv2^2+1\equiv 0 \pmod 5$

hence, $(n^2+1, 2n+1)=5$

Else, $5\not\mid(n-2)\implies (2n+1,n-2)=1$ and $5\not\mid(2n+1)\implies (n^2+1, 2n+1)=1$

Answer 3

Let $d=(a,b)$. Then $d| 4n^2+4n+1$ and $d|4n^2+4$ therefore $d|4n-3$. Since it also divides $4n+2$ we get $d|5$.

Answer 4

Let $d=\gcd(2n+1,n^2+1)$. Then since $d\mid 2n+1$ and $d\mid n^2+1$, we must also have $$d\mid(n^2+1)-(2n+1)=n(n-2)\;.$$ Clearly $\gcd(n,2n+1)=1$, and $d\mid 2n+1$, so $\gcd(n,d)=1$, and therefore $d\mid n-2$. By definition $d\mid 2n+1$, so $d\mid(2n+1)-2(n-2)=5$, and therefore $d=1$ or $d=5$.

The problem doesn’t require you to do so, but you can easily check that $d=5$ if and only if $n\equiv2\pmod 5$, so that $5\mid n-2$.

Answer 5

Clearly when $n=0$ the polynomials are relatively prime (gcd = 1).

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notice that $2n+1$ is always odd, while $n^2 + 1$ is always even. Thus the gcd can never contain $2$ as a factor.

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Let $d | n^2 + 1$ and $d | 2n + 1$, then [because $a|b$ implies $a|kb$] we have

• $d|2n^2 + 2$
• $d|2n^2 + n$

and [because $a|b$ and $a|c$ imply $a|c-b$] this gives us

• $d|n-2$

we can use this (since it's degree 1) to cancel out the $n$ in our assumption $d | 2n + 1$ to get

• $d|n + 3$

and again to get

• $d|5$

Since $5$ is prime, we've found that the GCD of the polynomials for some $n$ must be either $1$ or $5$.

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We would now like to find out exactly when the GCD is 5 (i.e. when $5|2n+1$ and $5|n^2+1$ and when it's one. For this we can use the equivalence [$a|b$ iff $b \equiv 0 \pmod a$]. So we only need to check 5 values.

n | 2n+1 | n^2+1
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0 | 1    | 1
1 | 3    | 2
2 | 0    | 0     !
3 | 2    | 0
4 | 4    | 2

therefore we have proved the result:

$$\gcd(2n+1,n^2+1) = \begin{cases}5 & \text{when } n \equiv 2 \mod 5 \\ 1 & \text{otherwise}\end{cases}$$