4
votes
2answers
341 views

Are there any patterns in simple continued fraction expansions of algebraic real numbers?

As we know there are patterns in simple continued fraction expansion of quadratic algebraic numbers,are there any patterns in simple continued fraction expansions of other algebraic real ...
3
votes
0answers
132 views

Algorithm to compute a common denominator of a finite set of rational numbers

Let $x_1,\dots,x_n \in \mathbb{Q} \cap (0,1)$ be fixed, but unknown, and assume that we know a number $K \in \mathbb{N}$ such that there is a number $N \in \{1,\dots,K\}$ such that $x_1 N,\dots,x_n N ...
7
votes
1answer
530 views

Can the Legendre symbol be calculated in polynomial time?

Is there an algorithm which on input "$(a,p)$" (where $0\leq a<p$ are integers) takes time polynomial in $\log p$ and outputs "NOT PRIME" if $p$ is not prime and otherwise outputs the Legendre ...
3
votes
2answers
126 views

All Integers from the Smallest Digit Stream with a Window Filter

Let's represent integers with D digits where each digit has B values (i.e., the base is B and we effectively work only with integers between 1 and B^D). Is it possible to choose a single ...
35
votes
1answer
1k views

improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...
2
votes
2answers
196 views

Integers up to n having an even number of trailing zeros in their factorial

It is well-known that the number of trailing zeros in the factorial $k!$ is given by the nice function $$ z(k) := \sum_{i \ge 1} \left\lfloor \frac{k}{5^i} \right\rfloor. $$ Now assume that we want ...
0
votes
0answers
74 views

Minimize the length of two disjoint segments in the string with given property

You are given a string s of size n, consisting of characters A and B only. You have to find minimum sum of size of the two disjoint segments of the string s such that number of A's in them are >= z. ...
3
votes
0answers
138 views

What is the function like when its Mobius inversion is $\sum_{w|r, (w,t)=1}\mu(w)q^{r/w}$?

Everyone, I am now reading a paper named The Irreducible Factors of $(cx+d)x^{q^m}-(ax+b)$ over $GF(q)$, http://qjmath.oxfordjournals.org/content/14/1/61.extract. And I’m confused with one of its ...
0
votes
0answers
65 views

approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...
2
votes
1answer
475 views

Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem: Exponentiating Polynomial Root Problem (EPRP) Let $p(x)$ be a polynomial with $\deg(p) \geq ...
8
votes
1answer
511 views

How to check whether a positive integer can be written as linear combination of given others, where all coefficients are positive?

Let $n$, $k$ and $m_1, \dots, m_k$ be positive integers. Which is the most efficient algorithm to find out whether there are positive integers $a_1, \dots, a_k$ such that $n = \sum_{i=1}^k a_i m_i$? ...
6
votes
4answers
490 views

How long does it take to compute a class number?

I was wondering if there are any known (upper and lower) bounds for the complexity of computing the class-number of a finite extension of the rationals. (A general bound should be in function of the ...
20
votes
4answers
1k views

Securing privacy of “who communicates with whom” under Orwell-like conditions

Assume that there is a big and powerful country with an information-greedy secret service which has backdoors to all internet nodes throughout the world which permit him to observe all exchanged data ...
3
votes
2answers
208 views

Complexity of a problem remotely related to the discrete logarithm $A=x g^x$

Let $x,g \in \mathbb{F}_p^\ast$. Given $g$ and either $$ A = x g^ x$$ or $$ A = x g^{x^2-1}$$ find $x$. What is the complexity of solving this? Is there a reduction to the discrete ...
2
votes
1answer
255 views

(efficient) method to test $\{n\alpha\}\not\in [A, B]\subset [0,1]$

Suppose $\alpha$ is a fixed given irrational number with $\alpha\in [A, B]\subset [0,1]$, are there any (efficient) methods to compute the least integer $n$ such that the decimal part of $n\alpha$ ...
9
votes
7answers
2k views

Is there an algorithm to solve quadratic Diophantine equations?

I was asked two questions related to Diophantine equations. Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all ...
2
votes
1answer
182 views

RefReq: Algorithms for standard operations in Algebraic Number theory

Given an algebraic number field $F$ (I actually don't have an idea how to implement this data already, except for splitting fields of polynomials, but there is something in SAGE) is there free code ...
28
votes
3answers
2k views

Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And ...
1
vote
1answer
135 views

What algorithms do you know for beltway reconstruction?

I've faced the beltway reconstruction problem and I've developed a simple backtrack algorithm, what algorithms do you know for this problem? Beltway Reconstruction Problem: Assume there is a set of ...
3
votes
1answer
208 views

Is the prime ideal principal which is in 256-th cyclotomic ring lying over 257?

As we known, the integer ring R of 256-th cyclotomic field is not a Principal Ideal Domain. And rational prime 257 is split completely in R. Suppose prime ideal P of R is an arbitrary ideal lying ...
3
votes
1answer
341 views

Fastest Digit Extraction for Any Irrational Number

I believe the current lowest-memory algorithm for computing the $n^{th}$ binary digit of $\pi$ requires $O(log(n))$ bytes and $O(n^2 log(n))$ days (I pick Bellard over Bailey–Borwein–Plouffe for ...
6
votes
2answers
727 views

Approximate number of primes below a given integer?

The problem of the complexity of the exact counting problem for primes is interesting. The best result we have about primes is that it is hard for TC0. But counting the number of witnesses to a TC0 ...
2
votes
1answer
322 views

At what point does Miller-Rabin become faster than trial division?

I've read in various places (and know) that Miller-Rabin is a much faster primality test than trial division for large $N$, but is much slower than trial division for small $N$. My question is: how ...
9
votes
2answers
594 views

When polynomial f(x^2) can be factored as g(x)·g(-x) ?

In relation to my question Expression for the sum of square roots of zeros of a polynomial How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where ...
0
votes
0answers
118 views

Find polynomial in finite field

We have $A$, $B \in GF(q^k)$ We want to find polynomial $h \in GF(q)[x]$ where $h(A) = B$ What is the lowest degree of $h$? How to find $h$ with the lowest degree and what is complexity of this ...
1
vote
1answer
284 views

Find root in finite field

What efficient algorithms exist for the solving $x^N = a$ in GF(q)? What are their complexities?
1
vote
1answer
271 views

calculate function from its divizor

There is elliptic curve $C (y^2 = x^3 + Ax + B)$ over $GF(q)$. There is algebraic function f on C. We have div(f). How calculate f as rational function ( $f = (f_1(x) + yf_2(x)) / (g_1(x) + ...
2
votes
0answers
189 views

Choosing a base where a given digit of a given number appears the most times

Is there an algorithm for choosing a base where a given digit of a given number appears the most times, that works better then trial and error? (see also this)
1
vote
1answer
253 views

Computing the ratio of two large integers modulo m

$P(n)$ and $D(n)$ are two large integers. Suppose $R(n) = \frac{P(n)}{D(n)}$ is an integer. I want to compute $R(n)\bmod m$. $P(n)$ and $D(n)$ are too large to be computed but $P(n)\bmod m$ and ...
1
vote
5answers
1k views

The Inverse of the Euler Totient Function

How can we calculate the cardinality of the inverse of Totient function of any positive integer n ? I tried going through this paper, but I couldn't understand the procedure. Thanks
2
votes
3answers
2k views

Algorithm for detecting prime powers

While reading Peter Shor's paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, I came across the following quote: "This scheme will thus work as ...
2
votes
3answers
275 views

Generating a set of integer passwords that can be securely authenticated

First, apologies for the title. This is an odd question, and I couldn't come up with a simple title for it. My question is as follows. Given a positive integer $k$, determine a set of properties ...
1
vote
2answers
773 views

The relationship between the Dirichlet Hyperbola Method, the prime counting function, and Mertens function

I have a question concerning the connection between the Dirichlet Hyperbola Method and properties of both the Mertens function and the prime counting function. Preliminary: Mertens function and the ...
9
votes
3answers
932 views

Mertens' function in time $O(\sqrt x)$

This MathOverflow question seems to indicate that the state of the art in computing $$ M(x)=\sum_{n\le x}\mu(n) $$ takes time $\Theta(n^{2/3}(\log\log n)^{1/3}),$ which matches my understanding. ...
3
votes
2answers
669 views

Find the maximum set whose subset sum is unique for every of its subset.

We are given a set of $n$ positive integers. We assume all of them are bounded by a polynomial of $n$. We would like to find a subset $S$ of numbers such that for any $T_1,T_2\subseteq S$, the sum of ...
6
votes
1answer
598 views

How this set of functions is ordered?

Notation: $k, m, n$ are non-negative integers $f, g, h$ are functions $\mathbb{N} \to \mathbb{N}$ $f^k$ is $k$-th iterate of the function $f$: $f^0(n)=n, f^{k+1}(n)=f^k(f(n))$ $f \prec g$ means ...
16
votes
5answers
3k views

Fastest Algorithm to Compute the Sum of Primes?

Can anyone help me with references to the current fastest algorithms for counting the exact sum of primes less than some number n? I'm specifically curious about the best case running times, of ...
5
votes
2answers
312 views

Feasibility of linear equations with few variables mod k

Say I want to verify the feasibility of $$Ax \equiv b \text{ (mod } k)$$ where $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m}$ and $k \in \mathbb{N}$. Is there a fast way to verify if there ...
2
votes
0answers
288 views

Algorithm for keeping a concrete version of Euclid's argument simple

(A version of this same question was posted to stackexchange.) Suppose we do what Euclid wrote about: starting with a finite set of primes, multiply them, add or subtract 1, factor the result, append ...
3
votes
1answer
395 views

Find the least prime $p$ such that $mn$ divides $p-1$

My hope is that this question is "trivial," but it is outside my knowledge base, so I'd appreciate some advice. Given positive integers $m$ and $n$, find the least prime $p$ such that $p-1 = mnk$ ...
1
vote
0answers
147 views

Finding a set of congruences from a set of solutions

Let $S_1, \ldots, S_m \in \mathbb{Z}_2^n$ be $n$-tuples forming a subgroup of $\mathbb{Z}_2^n$. How can I find a set of congruences (mod 2) over $x_1, \ldots, x_n$ satisfying exactly $x_1 = S_j[1], ...
0
votes
0answers
174 views

Number of biquadrates mod n

Is there an explicit formula for the number of fourth powers mod n? Finch & Sebah [1] give theorems, partially folklore, for squares and cubes mod n, but I don't know of a similar formula for ...
7
votes
2answers
602 views

Factoring some integer in the given interval

I'm posting this question here (rather than on CSTheory) since it seems to require much more knowledge about number theory than algorithms. Let N be a positive integer. Is there an efficient (i.e. ...
8
votes
1answer
660 views

The Number of Short Vectors in a Lattice

Given a lattice $L = \bigoplus_{i=1}^{m} \mathbb{Z}v_i$ (the $v_i$ are linearly independent vectors in $\mathbb{R}^n$) and a number $c > 0$, can one quickly compute or find a good estimate on the ...
3
votes
0answers
209 views

Implementation for computing Shintani domains

By "Shintani domain", I mean a fundamental domain for the action of the totally positive units of a totally real number field k with $[k \colon \mathbb{Q}]=n$ (or more generally, those congruent to 1 ...
11
votes
3answers
769 views

Bounds on squarefree numbers

Let $q_1,q_2,\ldots$ denote the squarefree integers 1, 2, 3, 5, .... What effective bounds are known for $q_n$? Clearly $$q_n\sim\zeta(2)n$$ but I need hard inequalities. Of course from the above ...
13
votes
2answers
897 views

Subfields of a function field

Is there an algorithm for generating (some or all) subfields of a certain genus of a given function field (even a random one,I mean for example generating a random elliptic subfield of a certain given ...
4
votes
1answer
344 views

Can we count primes in residue classes quickly?

Using combinatorial methods (due to Legendre, Lehmer, Meissel, Lagarias, Miller, Odlyzko, Deléglise, Rivat, and probably others) it's possible to count the number of primes up to $N$ quickly -- in ...
2
votes
5answers
1k views

partitioning a number into two sets based on sum of digits

hi, how can one determine whether a number belongs to such a set of numbers whose every element can be divided into two sets such that the sum of digits of each set of the number is same. E.g 23450 ...
6
votes
3answers
923 views

Prime counting - any fast alternatives to the Lagarias-Miller-Odlyzko combinatorial method or the Lagarias-Odlyzko analytical methods?

I guess the question says it all - I'm trying to track down fast algorithms for prime counting to know what's out there. I'm already familiar with the two algorithms mentioned in the title ...