Find the greatest common divisor of two consecutive terms of the sequence an=100+n^2.

Question

Looking at sequences a=u+n^2, that there is a common divisor of 4u+1 between a2u and a2u+1, and now i need to prove that this is the greatest common divisor between terms au and au+1. This is a part where i'm stuck, i know that au+1 - au is equal to 2n+1, but dont know where to go next?

Answer 1

Fix $n$ and let $d$ be the common divisor of $a_{n} = 100 + n^{2}$ and $a_{n+1}=100 + (n+1)^{2}$. Then $d$ divides their difference $$ a_{n+1} - a_{n} = 2n+1. $$ Furthermore, it divides the following linear combination: $$ 2a_{n} - n(2n+1) = 200 + 2n^{2} - 2n^{2} - n = 200 - n. $$ Again, $d$ divides the linear combination $$ 2(200 - n) + (2n + 1) = 400 + 1 = 401. $$ Since $401$ is prime, the only posibilities for $d$ are $1$ and $401$.

One can show that $401$ divides $100+n^{2}$ if and only if $n$ has residue $200$ or $201$ when divided by $401$. Then, the answer is: $$ d = \begin{cases} 401 \text{ if $n$ has residue $200$ or $201$ when divided by $401$,}\\ ~~1~~ \text{ in any other case}. \end{cases} $$