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Let $T colon mathbb{N}to mathbb{N}$ denote the $3x+1$ function, where $T(n)=n/2$ if $n$ is even, $T(n)=(3n+1)/2$ if $n$ is odd. As an accelerated version of $T$, we define a emph{jump} at $n ge 1$ by $textrm{jp}(n) = T^{(ell)}(n)$, where $ell$ is the number of digits of $n$ in base 2. We present computational and heuristic evidence leading to surprising conjectures. The boldest one states that for any $n ge 2^{150}$, at most four jumps starting from $n$ are needed to fall below $n$, a strong form of the Collatz conjecture.
We present two sets of 12 integers that have the same sets of 4-sums. The proof of the fact that a set of 12 numbers is uniquely determined by the set of its 4-sums published 50 years ago is wrong, and we demonstrate an incorrect calculation in it.
We describe a family of graphs with queue-number at most 4 but unbounded stack-number. This resolves open problems of Heath, Leighton and Rosenberg (1992) and Blankenship and Oporowski (1999).
This note is devoted to the investigation of Susskinds proposal(arXiv:1802.01198) concerning the correspondence between the operator growth in chaotic theories and the radial momenta of the particle falling in the AdS black hole. We study this propos
We prove that the Coleman-Mazur eigencurve is proper over the weight space for any prime p and tame level N.
By combining a sieve method of Harman with the work of Maynard and Tao we show that $$liminf_{nrightarrow infty}(p_{n+m}-p_n)ll exp(3.815m).$$