Related papers: Upper and lower estimates for integer complexity

Define $||n||$ to be the \emph{complexity} of $n$, which is the smallest number of $1$s needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $||n|| \geq 3\log_3 n$ for all $n$.…

Define $|n|$ to be the complexity of $n$, the smallest number of 1's needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $|n|\ge 3\log_3 n$ for all $n$. Define the defect of $n$,…

We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. $\left\| n \right\|$ denotes the minimum number of 1's in the expressions representing $n$. The logarithmic complexity $\left\|…

The complexity $f(n)$ of an integer was introduced in 1953 by Mahler & Popken: it is defined as the smallest number of $1$'s needed in conjunction with arbitrarily many +, * and parentheses to write an integer $n$ (for example, $f(6) \leq…

Define $||n||$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. The set $\mathscr{D}$ of defects, differences $\delta(n):=||n||-3\log_3 n$, is known…

The integer complexity $f(n)$ of a positive integer $n$ is defined as the minimum number of 1's needed to represent $n$, using additions, multiplications and parentheses. We present two simple and faster algorithms for computing the integer…

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$. Based on this, this…

Let $\Gamma$ denote the modular group $SL(2,\Bbb Z)$ and $C_n(\Gamma)$ the number of congruence subgroups of $\Gamma$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log C_n(\Gamma)}{(\log n)^2/\log\log n} =…

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$, leading this author and…

One of the major open problems in complexity theory is to demonstrate an explicit function which requires super logarithmic depth, a.k.a, the $\mathbf{P}$ versus $\mathbf{NC^1}$ problem. The current best depth lower bound is $(3-o(1))\cdot…

We investigate the number $R_3(n)$ of representations of $n$ as the sum plus the product of three positive integers. On average, $R_3(n)$ is $\frac{1}{2}\log^2 n$. We give an upper bound for $R_3(n)$ and an upper bound for the number of $n…

Let $k$ and $n$ be positive integers, $n>k$. Define $r(n,k)$ to be the minimum positive value of $$ |\sqrt{a_1} + ... + \sqrt{a_k} - \sqrt{b_1} - >... -\sqrt{b_k} | $$ where $ a_1, a_2, ..., a_k, b_1, b_2, ..., b_k $ are positive integers…

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. Define $n$ to be stable if for all $k\ge 0$, we have $\|3^k n\|=\|n\|+3k$. In [7],…

For a positive integer $n \geq 2$, define $t_n$ to be the smallest number such that the additive energy $E(A)$ of any subset $A \subset \{0,1,\cdots,n-1\}^d$ and any $d$ is at most $|A|^{t_n}$. Trivially we have $t_n \leq 3$ and $$ t_n \geq…

We prove that \Omega(n log(n)) comparisons are necessary for any quantum algorithm that sorts n numbers with high success probability and uses only comparisons. If no error is allowed, at least 0.110nlog_2(n) - 0.067n + O(1) comparisons…

"The Baron's omni-sequence", B(n), first defined by Khovanova and Lewis (2011), is a sequence that gives for each n the minimum number of weighings on balance scales that can verify the correct labeling of n identically-looking coins with…

Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by F\"{u}rer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg…

We consider the complexity for computing the approximate sum $a_1+a_2+...+a_n$ of a sorted list of numbers $a_1\le a_2\le ...\le a_n$. We show an algorithm that computes an $(1+\epsilon)$-approximation for the sum of a sorted list of…

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\| \ge 3\log_3 n$ for all $n$. Define the defect of…

We show that any quantum algorithm deciding whether an input function $f$ from $[n]$ to $[n]$ is 2-to-1 or almost 2-to-1 requires $\Theta(n)$ queries to $f$. The same lower bound holds for determining whether or not a function $f$ from…