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Results tagged with nt.number-theory
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user 3206
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
19
votes
3
answers
2k
views
Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?
Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, usin …
- 3,209
8
votes
How did Cole factor $2^{67}-1$ in 1903?
I imagine it took Cole longer than he said. If I were to undertake the project, here is how I would proceed:
I would start sifting the set of numbers {134k + 1} for primes. One can modify the Sieve …
- 3,209
6
votes
Accepted
A conjecture on the prime counting function
Look for a large gap in the distribution of primes. For this conjecture, the gap between $n!+2$ and $n!+n$ will suffice. Set $y = n!+2$ (which is composite) and set $m$ (which will be $\frac{x+y}{2} …
- 3,209
5
votes
Advice for number theory library
I have not seen mention of Władysław Narkiewicz (feel free to correct the accents and other marks) and his books on the history of number theory, as well as Ribenboim's texts on certain Diophantine eq …
4
votes
Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from...
The post above has a link to the term-complexity measure based on size of a term
computing a number. The following different model is from my memory of the BCSS paper, so verification would be apprec …
- 3,209
4
votes
What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt \cd...
Noting that the answers with many components involve few primes, I assume that any answer involves no primes (or denominators with prime factors) greater than 36. The sum of 1/i from m to n can be est …
- 3,209
3
votes
Accepted
Relative-totient function (2nd attempt)
I will not comment on the soundness of the approach, but I will render a subjective opinion: I don't like it. One of the reasons is that I have found most people do not have a good understanding of …
- 3,209
3
votes
How to attack this diophantine equation in 3 variables?
Note that x^3 - x = (x-1)x(x+1). Now let x = (a+b+c)
and rewrite the equation as
(x-1)x(x+1) = 987654320*a + 123456788*b.
Let D be gcd(987654320,123456788) = 16. There are
integers A, B so that 987 …
- 3,209
3
votes
Number of partitions of a number on a combinatorial bracelet
This is a transform which tightens the problem and asks the reader to consult the bracelet literature.
Consider first the question as specified with the additional proviso that the number of beads is …
- 3,209
3
votes
How many 0, 1 solutions would this system of underdetermined linear equations have?
It may be obvious to the casual observer, but it only just hit me recently that Hamiltonian cycle can be reduced to this problem, so of course the decision and counting problems are hard. I do not kn …
- 3,209
3
votes
Number of divisors of an integer of form 4n+1 and 4n+3
A faster way would involve looking at the prime factorization of the integer.
Let n be pq, where q contains all the 1 mod 4 prime factors of n, and q contains all the 3 mod 4 prime factors. Then the …
- 3,209
2
votes
Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive ...
EDIT 2015.07.15 I believe there is no such integer $x$. See below for the rest of the edit. END EDIT 2015.07.15
I had some initial thoughts which seemed promising. They do not lead to a proof, but …
- 3,209
2
votes
Accepted
Proofs needed for observations regarding prime-partitionable numbers
Let's assume a limited (and unproved) version of Linnik's theorem: There is a prime $q$ of the form $kp + 1$ for $k \leq (p-2)$ and $p$ a prime. Experimentally this is true, and can be proved for ma …
- 3,209
2
votes
Accepted
Results regarding the relative-totient function
As I understand the claim $\Lambda(x,n) = \frac{n'}{x'} \Phi(x) \pm V$, it is false for some $n$ and $x$ with $n$ close to $x$. Let us take $x$ to be $P_4=210$, the fourth primorial. Let us take $n$ …
- 3,209
2
votes
Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?
I've decided to simplify the argument found in notes of Jameson, and at the same time improve the bounds and ranges of applicability. I'm rewriting for the purpose of understanding and the specific g …
- 3,209