0
votes
1answer
95 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
0
votes
1answer
50 views

Parity of number of factors up to a bound?

Consider $b,n\in\mathbb{N}$ where $b\leq n$. We want to find the parity (ie. odd or even) of the number of divisors of $n$ that are $\leq b$. The question is to find a fast algorithm to find that ...
1
vote
2answers
30 views

MATLAB Newton non-linear equation

I have the following non-linear equation: where $w0=0.25,w0=0.5,w0=0.75$. I have to prove that if $k$ is a root, then also $−k$ is a root and that there exists only one $k∈(0,1)$ root, but my ...
1
vote
3answers
68 views

About the calculation of decimal digits of series up to the nth digit

Considering that we don't know any of the digits of some number defined as the limit up to infinity of a sum, I want to know how many terms do I have to sum to get the correct decimal representation, ...
2
votes
1answer
36 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
2
votes
0answers
51 views

Expected error due to the tablemakers' dilemma

[note: to me, this does not seem like a question for m.se, but on mathoverflow it has been retroactively closed, with very little indication of why or what might be corrected... and waiting for ...
3
votes
1answer
87 views

Fermat-quotient of “order” 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?

I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the Question for $$ b^{p-1} \equiv 1 \pmod{ p^m} \qquad \text{ with $p \in \mathbb P $, $1 \lt b \lt p$ and ...
0
votes
2answers
822 views

Checking the Harald Helfgott proof of the little Goldbach conjecture without a public release of numerical checks?

A few month ago, a proof of the little/ternary Goldbach conjecture has been claimed by Harald Helfgott with three articles: Major arcs for Goldbach's theorem Minor arcs for Goldbach's theorem ...
4
votes
4answers
203 views

How calculate $\pi$ to an accuracy of 10 decimal places?

Let $a=3.00000000001234...$ (irrational number) If $\overline{a}=3.00000000001$ (approximation $11$ places) then $|a-\overline{a}|<10^{-11}$ Note that the reciprocal is not satisfied: If ...
1
vote
1answer
70 views

Prime numbers and limit(?)

Can someone help me to prove the following: $$\lim_{x\to\infty}(\sum_{p\leq x}\frac{1}{p}-\log(\log(x)) -C)=0$$ Where $C$ is a proper constant. Thank you...
1
vote
0answers
88 views

gcd finding method

An integer $d$ is a $\gcd$ of two non-zero integers $a$ and $b$, if $d$ divides $a$ & $d$ divides $b$ '$c$ divides $a$ & $c$ divides $b$' implies '$c$ divides $d$' for any integer $c$. If ...
0
votes
1answer
67 views

How to normalize these numbers for better visualization?

My dataset is like this: a1 4565380 a2 676477 a3 359939 ... b1 222431 b2 12222 ... g1 139 ... h1 134 i1 10 j1 11 and goes on.. The problem is when I ...
82
votes
1answer
10k views

Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger

Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than 1, and ...
1
vote
0answers
100 views

Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

For some exercises with (divergent) summation of the Stieltjes constants I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as ...
1
vote
3answers
46 views

inequality with one number and a sum of numbers

Let $x_1, \ldots, x_n $ be non-zero real number, such that $\sqrt{x_1^2 + \cdots + x_n^2}=1$. Show that for any $i = 1, \ldots, n$, $$|x_i| \leq \sqrt{\frac{x_1^2 + \cdots + x_n^2}{n}}= ...
2
votes
0answers
75 views

A question on algorithm complexity

It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$ for a signal with bandwidth $N$. How to see or show that the fast Fourier transform ...
4
votes
1answer
91 views

Find $k$-tuples satisfies $j=n_2+2n_3+\cdots+(k-1)n_k$ if $n_1+\cdots+n_k=n$.

Let $n_i \in N$, $i=1,\ldots,k$ and such that $n_1+\cdots+n_k=n$. Fix $j \in N$. I would like to find all $k$-tuples (or algorithm how to find $k$-tuples) satisfies $$ j=n_2+2n_3+\cdots+(k-1)n_k $$ ...
13
votes
4answers
1k views

Detecting perfect squares faster than by extracting square root

Given the radix-$r$ representation of a integer $n$, and a small integer constant $k$, there is an $O(\log n)$ algorithm for detecting whether $n$ is a multiple of $k$, namely, division, which ...
0
votes
1answer
328 views

Calculating number of draws in a series of team matches

May be this could be an easy problem but somehow cannot arrive to any conclusive result to this indecisiveness problem For a certain number of teams ($n$) playing a series of matches ($m$), and given ...
1
vote
0answers
983 views

Teach me a simple, efficient division algorithm

I want to implement arbitrary-precision arithmetic in JavaScript for non-negative integer numbers. Long division isn't efficient if instead of the usual 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) there ...
23
votes
1answer
464 views

Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field

If $K$ is a number field, whose Galois closure over the rationals has degree 24 or so, and whose discriminant is around $163^4$, then what is a numerically efficient way of computing the first few ...
0
votes
2answers
57 views

a numerical concluding about (a/(a+b)) and (c/(c+b))

Let $a,b,c$ be three integers greater than $0$, and assume there is a real number $t$ such that $$ \frac{a}{a+b}=\frac{\left\lfloor t\right\rfloor}{\left\lfloor t\right\rfloor+1}. $$ Is there a way to ...
5
votes
0answers
128 views

Shintani cone zeta function

Is there a procedure/algorithm for calculating sums of the form $$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$ where $$ L_i(n_1,\ldots, n_r) ...
4
votes
1answer
520 views

Efficiently calculating the logarithmic integral with complex argument

My number theory library of choice doesn't implement the logarithmic integral for complex values. I thought that I might take a crack at coding it, but I thought I'd ask here first for algorithmic ...
3
votes
2answers
304 views

'(Pseudo)-random functions' by seeding of PRNGs?

I have an application that wants controllable random functions from $\mathbb{Z}^2$ and $\mathbb{Z}^3$ to $2^{32}$ , where by controllable I basically mean seedable by some parameters (say, on the ...
6
votes
8answers
4k views

Need faster division technique for $4$ digit numbers.

I have to divide $2860$ by $3186$. The question gives only $2$ minutes and that division is only half part of question. Now I can't possibly make that division in or less than $2$ minutes by applying ...