4
votes
1answer
330 views

Finding a rational number which is simply normal to relatively prime bases

Let $n\ge 2\in\mathbb Z$. Suppose that a base-$n$-decimal $(0.a_1a_2a_3\cdots)_n$ represents $\sum_{k=1}^{\infty}\frac{a_k}{n^k}$ where $a_{i}\in\{0,1,\cdots,n-1\}\ (i=1,2,\cdots)$ is each digit ...
0
votes
1answer
24 views

Binary expansion

I am trying to get my head around the left and right shift for binary expansion. The rules are: Shifting to the right ...
0
votes
0answers
11 views

how to write a lattice $[\alpha,\beta ]$ in the form [$a,b+c\omega _7$]

$\fbox{1}$ if we write [$2-\sqrt{7},5+3\sqrt{7}$] in the form [ $a,b+c\omega _7$],what is the value of $a,b,c$ $\omega=\sqrt{7}$,since $ 7\equiv 3\mod 4$ $N(2-\sqrt{7})=4-7=-3$ $N( ...
1
vote
0answers
83 views

conversion of number from base 10 to base 16

I have trouble finding an example of a conversion of a number from base 10 to base 16, where there is a loss of significant digits when using the Euclidean algorithm due to division by large numbers, ...
3
votes
1answer
101 views

Lower bound on a number theoretic function

Let $n$ be a positive odd integer, let $$n_j = \Bigl\{\frac{n}{2^{j+1}}\Bigr\}\,,$$ where $\{x\}$ denotes the fractional part of $x$, and finally let $k = \lceil \log_2 n\rceil$. Consider the ...
5
votes
1answer
162 views

Efficient division using binary math

I'm writing code for an FPGA and I need to divide a number by $1.024$. I could use floating and/or fixed point and instantiate a multiplier but I would like to see if I could do this multiplication ...
1
vote
1answer
145 views

Find the largest divisor of an integer $b$.

I want to find out an efficient method to calculate the largest divisor of a very big integer $b$ which can be up to $\large 2^{1000}$. That is, I want to find out an integer $a < b$, such that ...
3
votes
2answers
75 views

How do we know $p/q$ can be expressed as a terminating fraction in base $B$ only if prime factors of $q$ are prime factors of $B$?

On cs.stackexchange I asked a math question: How to demonstrate only 4 numbers between two integers are multiples of .01 and also writable as binary. Yuval Filmus answered with a explanation ...
1
vote
0answers
123 views

Finding the maximum XOR metric

I'm trying to find a way to find n keys (x bits) where the XOR distance metric between them would be greatest. By XOR distance metric I just mean the value when two keys are XORed together. So for ...
1
vote
2answers
63 views

Special binary string

Imagine a binary string of increasing length, up to infinity. What makes it so special? Well, just a simple "rule": for any given length (odd or even), if one folds the string in half, there is at ...
1
vote
1answer
113 views

XOR for 10 and 20

I know that this is the XOR truth table. A B Q ------ 0 0 0 0 1 1 1 0 1 1 1 0 I have a = 10; and b=20; Their respective binaries are a=1010; and b=10100; a ...
5
votes
2answers
241 views

Nim addition- binary addition without carrying

A nim addition table is essentially created by putting, in any cell, the smallest number not to the left of the cell and not above that cell in its column. However, I know for a fact that nim addition ...
3
votes
0answers
652 views

Binary representation of powers of 3

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this ...
3
votes
3answers
1k views

A proof that powers of two cannot be expressed as the sum of multiple consecutive positive integers that uses binary representations?

In this earlier question, the OP asks for a proof of the statement Every natural number not of the form $2^k$ for some natural number k can be written as the sum of two or more consecutive ...