# Tagged Questions

This tag concerns computational problems central to mathematical and scientific computating. The scope includes algorithms, numerical analysis, optimization, and linear algebra, computational topology, computational geometry, symbolic methods, and inverse problems.

44 views

32 views

### n-th number with given prime divisors

I would like to compute th $n$-th positive integer whose prime divisors are among numbers $2$, $3$ and $5$. $n$ is at most $12500$. My first approach was sieving but i found out that there are less ...
218 views

### How many numbers $N \le 10^{10}$ are the product of $3$ distinct primes?

How many numbers $N \le10^{10}$ are the product of $3$ distinct primes? I can realistically calculate any $\pi(n), n < 10^{15}$ but I don't think it's possible to list all primes $>10^8$ in ...
24 views

### Are there alternatives to polygons in mathematical (computational) modelling?

So polygons are pretty standard in computer graphics, but from a mathematical perspective, one'd expect something more refined and sophisticated to be possible right? Polygons are not very ...
10 views

### Understanding the bound given by Johnson–Lindenstrauss lemma

Here I choose to use the statement made by S.Dasgupta: For any $0<\epsilon<1$ and any integer $n$, let $k$ be a positive integer s.t. $$k \geq 4(\epsilon^2/2-\epsilon^3/3)^{-1} \ln n$$ Then ...
29 views

### Egyptian fraction with least possible sum

Suppose that $~a~$ and $~b~$ are coprime positive integers. Then there exists representation of $~\frac{a}{b}~$ as egyptian fraction: $$~\frac{a}{b} = \frac{1}{d_1} + \cdots + \frac{1}{d_s} ~$$ There ...
24 views

17 views

### Bezout coefficients with least absolute sum

Let $a$ and $b$ be some integers and $d$ is their gcd. By Bezout's identity there exist such $x$ and $y$ that $ax+by=d$. I wonder when sum of absolute values of $x$ and $y$ is minimal? I'm ...
34 views

### How to Solve an Integral Equation for an Unknown Integrand numericlaly?

I am working on an astrophysical research in which we relate the cumulative number of Damped Lyman Alpha HI clouds/galaxies, namely their number densities, $\frac{dN_{DLA}}{dz}(>M, z=0),$ to the ...
252 views

### Playing with Fermat's little theorem

Fermat's little theorem states that if $~p~$ is a prime number then for any integer $~a~$ the number $~a^p - a~$ is divisible by $~p~$. What if one fixes the exponent $~n~$ and tries to find all $~m~$...
43 views

### Proper code for $p\to q$ in GAP

I’d like to know if gap> IsBool(not(p) or q)=true; is the only code for checking the trueness of a conditional statement ...
33 views

### Using only postage stamps of value 64 and 55, how can I work out the way to get closest to a high parcel value?

Searching has shown many questions like this for values of 4 and 7 cents, but nothing for higher values. For British postage, first class stamps are £0.64 and second class are £0.55. Low value stamps ...
18 views

### In the Miller–Rabin primality condition why do we know the odd power case is congruent to 1?

A prime number n satisfies $a^{d} \equiv 1\pmod{n}$ or $a^{2^r\cdot d} \equiv -1\pmod{n}$ where $n - 1 = 2^s·d$, d is odd and $r = 0, 1, ..., s-1$ Why does the second condition being false imply ...
167 views

### What exactly are the numbers we use everyday?

Pi can be defined as diameter / circunference of a circle. But what is a circle? You can't tell a computer: "build a circle and divide its diameter by its ...
36 views

### Harshad numbers with given sum

By definition Harshad number for base $~10~$ is any number divisible by sum of its decimal digits. Wikipedia gives some information on such numbers but i still have some questions and unforunately i ...
23 views

46 views

101 views

### Largest prime gap under $2^{64}$

Thanks to Tomás Oliveira e Silva's extensive calculations, it is known that the largest prime gap less than $4\cdot10^{18}\approx2^{61.8}$ is 1476. I'd like an upper bound for the largest prime gap ...
58 views

### Instability of Kuramoto solutions with Runge Kutta 4th order method

I am currently trying to solve the Kuramoto model: $\ddot{\theta_i} = P_i - \alpha\dot{\theta_i} + K \underset{i \neq j}{\sum}\sin(\theta_i - \theta_j)$ I split this second order differential ...
24 views

### Help with Legendre Plot Matlab

I've written a code to change a Chebyshev into a Legendre Polynomial, however it will not plot the graph after and I'm not sure why the graph will not plot? The code i have is: function LegendrePoly(n)...
105 views

### How to calculate $10^{0.4}$ without using calculator

How to calculate $10^{0.4}$ without using calculator or if not what is the closest answer you can get just using pen and paper within say $2$ min?
35 views

### Computing midpoint of an interval overflow

For computing the midpoint m of an interval $[a, b]$, which of the following two formulas is preferable in floating-point arithmetic? Why? When? (Hint: Devise examples for which the "midpoint" given ...
14 views

### Equivalence and interoperability of computation systems using a calculus

I am trying to prove that two computational systems are interoperable and both can be converted into another parent system . So computational system A,B can be mapped into computational system C. I ...
52 views

### Cannot understand solution (Turing Machine & Reduction)

Photo of my problem that I don't understand About question above in photo, I just can't understand its solution provided. We know the complement of Atm = {...
19 views

### Method to Linearise PDE

I have a Monge-Ampere-type PDE I wish to solve using a finite difference method: $$(1-u_{xx})(1-u_{yy}) -u_{xy}^2 = f(x,y).$$ Is there generally a preferred method for linearising the system after ...
22 views

### Different methodology for maximizing entropy in continuous random variable case

Suppose we want to maximize the well-known Shannon entropy $S=-∫_{0}^{x_{max}}f(x)lnf(x)dx$ subject to the following constraints $∫_{0}^{x_{max}}f(x)dx=1$, $∫_{0}^{x_{max}}xf(x)dx=x ̅$ and so on (...