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Results tagged with nt.number-theory
Search options questions only
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user 5712
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
6
votes
1
answer
585
views
Equal digit sums
Let $s(a)$ be the sum of decimal digits of a number $a$. Is it known that for any $a\ne b$ exist $n$ such that $s(na)\ne s(nb)$?
- 11.5k
3
votes
1
answer
417
views
The digit sums again: how far they can be equal?
As we started "must-to-ask" questions, see Seva's post we should ask one more natural question.
For integer $n\ge 0$, let $s(n)$ denote the sum of the digits in the decimal representation of $n$.
1 …
- 11.5k
3
votes
0
answers
255
views
Cassels' algorithm vs. “divided cells” algorithm
Cassels' algorithm mentioned in link text looks similar to Delone's “divided cells” algorithm. Are there any differences in these algorithms?
- 11.5k
2
votes
2
answers
1k
views
Solve in positive integers $n!=m^2$
Is anybody know a solution of this problem?
(Sorry, correct question is here.)
- 11.5k
5
votes
1
answer
889
views
On a sum involving Euler totient function
Let
$$S_a(N)=\sum_{n\le N}\frac{\varphi(an)}{n^2}.$$
The usual machinery gives an asymptotic formula
$$S_a(N)=\frac1{\zeta(2)}\cdot\frac{a^2}{\varphi_+(a)}\log N+C(a)+O(N^{-1+\varepsilon}a^{1+\varepsi …
- 11.5k
35
votes
3
answers
4k
views
Solve in positive integers: $n!=m(m+1)$
Does anybody know a solution to this problem? (Sorry, I've missed one summand in the previous post.)
- 11.5k
12
votes
1
answer
390
views
Lengths of continued fractions for the numbers with fixed ratio
Let $s(x)$ is the length of continued fraction expansion of $x$, and let $l(x)$ be the sum of partial quotients. I can prove that for any rational $\alpha$ ratios $\frac{s(\alpha x)}{s(x)}$ and $\frac …
- 11.5k
3
votes
1
answer
228
views
Addition law for elliptic curves of the form $x^2y^2+a(x+y)+b=0$
Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?
- 11.5k
4
votes
1
answer
130
views
Calculation of one constant similar to MZV
The series arose in the calculation of Mean value of a function associated with continued fractions:
$$C=\sum_{1\le b\le d<\infty}\frac{1}{b(b+d)d^2}.$$
Obviously
$C=C_1-C_2,$
where
$$C_1=\sum_{1\le …
- 11.5k
11
votes
1
answer
388
views
Is Somos-8 $\mod 2$ periodic?
It is known that the Somos-$k$ sequences for $k\ge 8$ do not give integers. But the first terms of Somos-8 sequence $s_n=a_n/b_n$
$$1, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 25, 61, 187, 775, 5827, 14815,\fra …
- 11.5k
7
votes
0
answers
327
views
$n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test
It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of divis …
- 11.5k
32
votes
1
answer
957
views
Is there a regular pentagon with a rational point on each edge?
This question was asked by Yaakov Baruch in the comments to the question Can a regular icosahedron contain a rational point on each face? It seems that this question deserves special attention.
- 11.5k
2
votes
1
answer
246
views
The number of different lattice triangles
Two convex lattice polygons are equivalent if there is a lattice-preserving affine transformation mapping one of them to the other. Equivalent polygons have the same area. Let $H(A)$ denote the number …
- 11.5k
9
votes
0
answers
227
views
De Bruijn sequence inside De Bruijn sequence
A binary De Bruijn sequence of index $n$ is a circular sequence $S=a_1a_2\ldots a_{2^n}$, with $a_i∈\{0,1\}$, and such that each of the $2^n$ binary $n$-tuples occurs exactly once in $S$.
What is …
- 11.5k
4
votes
1
answer
518
views
Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$
Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
EDT 1: A possible answer is Analysis of the subtr …
- 11.5k