# Questions tagged [comp-number-theory]

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### Least-buffer 2ᵏ→n∈⟨1;2ᵏ⟩∩ℕ base conversion of equal-probability messages source

Having an infinite source of messages of $2^k$ possible values of equal probability of occurrence, what is the least necessary buffer needed for a reversible conversion of it to a source of messages ...
424 views

### Is prime-counting function #P-complete?

Recall $\pi(n)$ the number of primes $\le n$ is the prime-counting function. By "PRIMES in P", computing $\pi(n)$ is in #P. Is the problem #P-complete? Or, perhaps, there is a complexity reason to ...
1k views

### Deciding whether an interval contains a prime number

What is the complexity of deciding whether an interval of the natural numbers contains a prime? A variant of the Sieve of Eratosthenes gives an $\tilde O(L)$ algorithm, where $L$ is the length of the ...
95 views

### Fixed dimension Integer programming minus LLL in fixed parameter $NC$?

If you remove LLL part then is remaining part of a. Lenstra algorithm b. Barvinok algorithm in $O(f(n)(\log(mL))^c)$ time on $O(g(n)(mL)^c)$ processors with fixed $c>0$ in fixed $n$...
75 views

### How to charactorize computational complexity based on finding solution to algebraic equations? [closed]

The Matiyasevich/MRDP Theorem relates two notions:one from computability theory, the other from number theory, thus Turing Machine and algorithms finding integral solution to algebraic equations can ...
206 views

### Relation between transcendental numbers and computational complexity?

Regarding the relation, there is the Hartmanis-Stearns conjecture, but beyond Turing Machine of the realtime output, there is no further conjecture or theorem. Obviously irrational algebraic number ...
286 views

### Is there any algorithm outputing $e$ in real time?

The Hartmanis-Stearns Conjecture says that a number computed in real time by a Turing Machine is either rational or transcendental. We know that there is some transcendental (Liouville) number that ...
182 views

### What is the computational complexity of Hilbert Tenth problem over $\mathbb{Q_p}$

We know that Hilbert Tenth problem over $\mathbb{Q_p}$ is decidable,so what is the computational complexity of Hilbert Tenth problem over $\mathbb{Q_p}$? Is it equivalent to Tarski elimination ...
309 views

### Algorithm to compute distance between powers

Given coprime $a, b$, can you quickly compute $$\min_{x, y > 0} |a^x - b^y|$$ Here $x, y$ are integers. Obviously taking $x = y = 0$ gives an uninteresting answer; in general how close can these ...
159 views

219 views

### Does a polynomial-time algorithm for factoring product of two primes imply a polynomial-time algorithm for factoring in general?

Is it known if the existence of a polynomial-time algorithm for the promise problem of factoring of numbers with two prime factors implies that factoring in general has a polynomial-time algorithm?
160 views

### Simplified lattices

Consider the following question: Let $N$ be some large prime number, and suppose we are given $n$ uniformly independent samples $g_i$ from $0...,N-1$. Think of $N$ as being exponentially large in $n$...
495 views

### Computational complexity of modular power towers (tetration)

The complexity of modular addition is known: $g + p \mod N$ (for $|p| \approx |g| \approx |N|$) can be computed in $O(n = |N|)$. The complexity of modular multiplication is open though some results ...
241 views