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3
votes
0answers
89 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
5
votes
1answer
259 views

Identity for frequency of integers with smallest prime(n) divisor

An identity for A038110 and A038111: $$ \frac{\phi(e^{\psi(p_{n}-1)})}{e^{\psi(p_{n})}}=\frac{\prod _k^{n-1} \left(1-\frac{1}{p_k}\right)}{p_n}, $$ where $\psi(\cdot)$ is the second Chebyshev function ...
4
votes
1answer
63 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
2
votes
0answers
34 views

Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)

I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm. I know that Maple has a PSLQ implementation, but ...
3
votes
2answers
49 views

Maximum likelihood with Bernoulli trials: what to do if there are no successes?

I'm analyzing the security of a secret sharing scheme. One attempt I'm analyzing is "blind luck". I return a random share and hope that noone notices. The probability $p$ of someone not noticing ...
0
votes
1answer
16 views

Types of graphs for this

How many ways could this be graphed, in a way such that it shows that the degree or slope of a surface affects the average speed. I think T1, t2 etc means at second 1, second 2. This was done using a ...
1
vote
1answer
40 views

Sum of Digit squares

I like to create problems for myself, While playing with my calculator, I found the following result. Take any positive digit $a_0$, and let $a_n$ denote the sum of digit squares of $a_{n-1}$. Then ...
3
votes
2answers
220 views

Writing computer code in a mathematics paper.

I have some computer computations (using GAP) in my research which I would like to write in the paper. How should I go about it? Should I include the full working code or just the major algorithms? ...
1
vote
0answers
32 views

The optimal way of improving fluency in elementary maths techniques, whilst holding a 9-5 job?

As an example of someone who has discovered maths at a later point in my life than average, and who has (perhaps unusually?) proven the point that it is perfectly possible to study undergraduate ...
1
vote
2answers
134 views

How many digits do we need for a proof ??

In the question: Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$, the value of that integral was ...
1
vote
2answers
206 views

In simple terms what is the sequence 220,171700,167167000,166716670000 a part of?

I know that <1,2,3,...,10>$\cdot$<1 0,9,8,...1>=220 <1,2,3,...,100>$\cdot$<100,99,98,...,1>=171700 <1,2,3,...,1000>$\cdot$<1000,999,998,...,1>=167167000 ...
2
votes
0answers
36 views

Book on higher-power trigonometric equation simplification techniques

Recently, I have become fascinated about learning techniques for simplifying experimentally derived trigonometric mathematical models that are raised to higher powers. Are there any good references ...
1
vote
0answers
16 views

How do I state a reduction in cost?

I have developed an algorithm and am having a hard time stating its benefit versus a baseline. Suppose that the baseline cost of solving the problem is 1000 seconds. Now suppose that my algorithm ...
3
votes
1answer
110 views

Interesting pattern that I found while finding dice that always roll a prime number

The original puzzle I was trying to solve is to find unique numbers to write on the faces of two dice such that the numbers you see on each dice after rolling sum to a prime. I realized that for this ...
2
votes
1answer
80 views

Unexpected data for primes?

I found some unexpected data for primes. Consider $p(n)$ being the product over all primes smaller than or equal to $n$. When factoring $p(n)^a +1$ for $a=1$ or $a=2$ we get the expected amount of ...
12
votes
1answer
255 views

Does $\pi$ satisfy the law of the iterated logarithm?

It is widely conjectured that $\pi$ is normal in base $2$. But what about the law of the iterated logarithm? Namely, if $x_n$ is the $n$th binary digit of $\pi$, does it seem likely (from ...
3
votes
1answer
116 views

Probability sequential terms of a linear congruential generator are coprime

I was following section 3.1.2 of the Structure and Interpretation of Computer Programming where they calculate $\pi$ using Cesàro's theorem that the probability two randomly chosen numbers are coprime ...
1
vote
1answer
113 views

Help with a different approach to extracting a polynomial equation from differences

It is well known that we can determine the degree of a polynomial can be found by finding when the differences are the same. i.e. if the second differences are the same, it is a polynomial of the 2nd ...
1
vote
1answer
47 views

Explanation over this simple equation, which involves multiplication of two square-roots

It was the time in schools, where we were taught about finding square roots of numbers. For a long time, i found the technique very difficult, so always used to find some shortcut to get rid of the ...
-1
votes
4answers
430 views

Divide by a number without dividing.

Can anyone come up with a way to divide any given x by any given y without actually dividing? For example to add any given x to any given y without adding you would just do: $x-(-y)$ And to ...
20
votes
9answers
788 views

Free online mathematical software

What are the best free user-friendly alternatives to Mathematica and Maple available online? I used Magma online calculator a few times for computational algebra issues, and was very much satisfied, ...
9
votes
5answers
320 views

Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$

Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed: ...
45
votes
2answers
1k views

Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, ...
1
vote
2answers
72 views

How to find length when viewing at some angle?

I have a question on angles. I have a rectangular tile. when looking straight I can find the width of the tile, but how do I find the apparent width when I see the same rectangular tile at some ...
0
votes
1answer
23 views

look for a metric for a two variable system

I have a series of experiments for different objects from the experiments let me put it in an abstract way there is a condition A for a specific object O the success rate/percentage is p(A) generally ...
0
votes
1answer
64 views

simplifying “$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}$”

Is this equality correct? For finite sets $A$ and $B_a$ (where $a\in A$), we have: $$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}=\sum_{f\in \prod_{a\in A}B_a}\quad \prod_{a\in A}{h(a,f(a))}$$
0
votes
3answers
121 views

Given $n+1\mid2\sum_{k=1}^{n}{a_k}$, find $a_k$.

Let $m$ be a positive integer. There are only 2 finite sequences of positive integers like $a_1,a_2,...,a_m$ such that $$(\forall n \leq m)\left(n+1\mid2\sum_{k=1}^{n}{a_k}, \quad a_n\in [1,m],\quad ...
4
votes
2answers
148 views

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png An equilateral triangle can be divided into 3 congruent shapes. Questions: ...
13
votes
1answer
312 views

Yet another nested radical

Consider $$F(x) = \sqrt{x -\sqrt{2x - \sqrt{3x - \cdots}}}$$ I believe I can prove (with some handwaving) that $F$ does converge everywhere in $\mathbb{C}$ $\Im F = 0$ for sufficiently large real ...
2
votes
0answers
99 views

What would be the most useful BOINC project to help researchers?

There are some distributed computing projects like BOINC? What would be the most useful BOINC project to help mathematicians or are those just waste of energy and time?
35
votes
10answers
2k views

Mind-blowing mathematics experiments

We've all heard of some mind-blowing phenomena involving the sciences, such as the double-slit experiment. I was wondering if there are similair experiments or phenomena which seem very ...
4
votes
3answers
205 views

Probability of Random number selection

Suppose you are asked to pick any random real number. Then you have a choice to pick any number between -∞ and +∞, i.e, infinite numbers. The probability that you select a particular number n = 1/∞ = ...
0
votes
0answers
143 views

Tanh-Sinh integration in 2-dimensions.

I recently implemented a Tanh-Sinh quadrature integrator for a two dimensional integral, simply by integration first over the the first variable and then over the second. My question is whether or not ...
7
votes
2answers
659 views

Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? [closed]

Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? Experimenting a bit I also found $\zeta(\frac{8}{3}) \approx e^\frac{1}{4}$, $\zeta(\frac{31}{9}) \approx e^\frac{1}{8}$ and ...
2
votes
1answer
179 views

number of additive partition

I have a question related with number of additive partition or method similar like this: $$p(5)=1+4=2+3=1+1+1+1+1=1+1+1+2=1+2+2=1+1+3$$ For a given number $n$,if we are trying to calculate ...
8
votes
2answers
251 views

What experimental-mathematical problem would you try to solve if you had a supercomputer?

At the present moment, what open mathematical problem do you seriously think you could solve if you had a very powerful computer at your disposition? I mean something like the Four-Color Problem, i.e. ...
0
votes
3answers
1k views

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) ...
5
votes
1answer
166 views

Is there an explanation for the patterns formed by the binomial coefficients C(n+d-1,d) mod 512?

Simplicial sequences generalize the familiar "linear", "triangular", and "tetrahedral" number sequences. (A line segment is a 1-simplex, a triangle is a 2-simplex, a tetrahedron is a 3-simplex, and so ...
18
votes
3answers
530 views

Is there a real number lookup algorithm or service?

Is there a way of taking a number known to limited precision (e.g. $1.644934$) and finding out an "interesting" real number (e.g. $\displaystyle\frac{\pi^2}{6}$) that's close to it? I'm thinking of ...