Number Theory » Combinatorial N.T.

The 3n+1 conjecture ★★★

Author(s): Collatz

Conjecture   Let $ f(n) = 3n+1 $ if $ n $ is odd and $ \frac{n}{2} $ if $ n $ is even. Let $ f(1) = 1 $. Assume we start with some number $ n $ and repeatedly take the $ f $ of the current number. Prove that no matter what the initial number is we eventually reach $ 1 $.

Keywords: integer sequence

Posted by dododododo
updated July 11th, 2014
4 comments

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Combinatorics

Long rainbow arithmetic progressions ★★

Author(s): Fox; Jungic; Mahdian; Nesetril; Radoicic

For $ k\in \mathbb{N} $ let $ T_k $ denote the minimal number $ t\in \mathbb{N} $ such that there is a rainbow $ AP(k) $ in every equinumerous $ t $-coloring of $ \{ 1,2,\ldots ,tn\} $ for every $ n\in \mathbb{N} $

Conjecture   For all $ k\geq 3 $, $ T_k=\Theta (k^2) $.

Keywords: arithmetic progression; rainbow

Posted by vjungic
updated May 12th, 2010
1 comment