Problem

Does the sequence of squares contain an infinite arithmetic subsequence?

Solution - by Jo Hilsmann (Germany)

The difference between two consecutive squares is $(n+1)² - n² = 2n+1$, which is a strictly increasing integer sequence. Therefor an arithmetic sequence with stepsize $K$ must stop latest at the $N$-th term, when $2N+1$ is bigger than $K$. So no infinite arithmetic subsequence contained in the sequence of squares.

Thank you!

Huge thanks to B Sashi Kanth (Hyderabad, India), Arkaprovo Das (Kolkata, India), Kilian Väth (Canada/Germany), Ben D (New York), pj, Jason Wild (Germany), Kurt Wynn (Brockton Bay), Anubhab Santra (India), Júlio Dias Saraiva Viana (Salvador, Brasil), Bethany Epstein (Texas), Ammar Ratnani (Stanford), Matt (UIUC), jc (new york), Jiayi Wang(Shanghai), Brandon Howe (New Jersey), Jake Sun (Boston), and Jo Hilsmann (Germany) for submitting solutions to this challenge problem.