nt.number theory – Is almost every number the sum of two numbers with small radicals?

Define a set of numbers with small radicals (A341645 in OEIS) by
$$A_2={ninmathbb{N} ;|; text{rad}(n)^2le n}$$

The asymptotic density of $A_2cap {1,dots N}$ is $sqrt{N}times e^{2(1+o(1))sqrt{log N / log log N}}$, as per Lucia’s answer here.

  1. Main question: does the sumset $A=A_2+A_2$ contain all sufficiently large integers? In other words, is $mathbb{N}setminus A$ finite?

The number of misses is initially large but becomes sparse very rapidly. I didn’t find any after $86931723$, up to $10^9$. $A$ is not in OEIS (its complement is strictly a subset of A085253 there).

Other questions:

  1. for any prime $p$, do the elements not divisible by $p$ have relative asymptotic density $0$ in $A_2$?

  2. Computing (up to $10^9$) the subset $Bsubset A$ of sums of coprime pairs in $A_2$, points to the misses thinning out very slowly (still above $13%$ near $10^9$). Are there euristic arguments for or against $mathbb{N}setminus B$ being finite?

Is anything else known, or worth asking, about $A_2$, $A$ and $B$?

rt.representation theory – Question about a paper on radicals of module categories

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by
CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper:

enter image description here

Definition: Let $f: X rightarrow Y$ be an irreducible morphism in mod$(A)$, with $X$ or $Y$ indecomposable. The left degree of $f$ is infinite, if for each integer $n geq 1$, each module $Z in$ ind$(A)$ and each morphism $g: Z rightarrow X$ with dp$(g)=n$ we have that $fg notin$ rad$^{n+2}$(Z,Y).

Sadly I don’t understand the underlines part. How does it follow that the left degree of $g_1$ is infinite? Any help is appreciated!

simplifying expressions – force maximum simplification of radicals in a traditional way-How to do it?

this should be simple but I can’t get mathematica to simplify a radical or a set of radicals and express them in the traditional way
for example by simplifying this

$$text{FullSimplify}left(left(3 a sqrt(4){18 a^4}right)^3,a>0right)// TraditionalForm$$
results in this
$$81 2^{3/4} sqrt{3} a^6$$
or this
$$27 18^{3/4} a^6$$

but not this,

$$81 sqrt(4){72} a^6$$

which is the final and most compact

There is some way I can simplify even more and from the results in a traditional way, this is just a small example.

radicals – $frac{sqrt{2}+sqrt{6}}{sqrt{2}+sqrt{3}}$ is it a rational number?

I need to demonstrate the rationality of this number:

$$frac{sqrt{2}+sqrt{6}}{sqrt{2}+sqrt{3}}$$

I have tried using squares, the p/q definition of rationality and the facts that

1)rationalĂ— irrational=irrational (unless rational=0),

2)rational+irrational=irrational.

However, I haven’t been able to reach some conclusion.Any help would be appreciated!

Power series – complex nested radicals $ { Re} Big ( sqrt {1+ frac {i} {2} sqrt {1+ frac {i} {2 ^ 2} sqrt {1+ frac {i } {2 ^ 3} sqrt {1+ frac {i} {2 ^ 4} sqrt { cdots}}}} Big) = 1 $

One last question about nested radicals, but this time with a complex value:

$$ { Re} Big ( sqrt {1+ frac {i} {2} sqrt {1+ frac {i} {2 ^ 2} sqrt {1+ frac {i} {2 ^ 3} sqrt {1+ frac {i} {2 ^ 4} sqrt { cdots}}}} Big) = 1 $$

I tried to use power sets like Somos' answer, but I fail.

We also have a nice relationship
$$ S = { Re} Big ( sqrt {1+ frac {i} {2} sqrt {1+ frac {i} {2 ^ 2} sqrt {1+ frac {i} { 2 ^ 3} sqrt {1+ frac {i} {2 ^ 4} sqrt { cdots}}}} Big) + { Im} Big ( sqrt {1+ frac {i} { 2} sqrt {1+ frac {i} {2 ^ 2} sqrt {1+ frac {i} {2 ^ 3} sqrt {1+ frac {i} {2 ^ 4} sqrt { cdots}}}} Big) = sqrt {1+ frac {1} {2} sqrt {1+ frac {1} {2 ^ 2} sqrt {1+ frac {1} {2 ^ 3} sqrt {1+ frac {1} {2 ^ 4} sqrt { cdots}}}} = 1.25 $$
I also find it good to know what happened to Herschfeld's theorem in the case of a complex number. If someone has some paper on it, it would be cool.

My question: How do you solve it?

Thank you in advance .

Calculus – proof of a closed form for an integral with nested radicals

Is there an easy way to prove the following identity?

$$ int_0 ^ 1 sqrt { frac {u ^ 2-2-2 sqrt {u ^ 4-u ^ 2 + 1}} {4 u ^ 6-8 u ^ 4 + 8 u ^ 2-4 }} mathrm du = frac { sqrt {3 + 2 sqrt {3}}} {2 ^ {10/3} pi} Gamma left ( frac13 right) ^ 3 $$


context::

This integral was created when trying to evaluate the full elliptic integral of the first kind $ K (m) $ ($ m $ is the parameter),

$$ K left ( exp left ( frac {i pi} {3} right) right) $$

in terms of simpler functions. Specially,

$$ K left ( exp left ( frac {i pi} {3} right) right) = C left (1 + i left (2- sqrt {3} right) right ) $$

and $ C $ is the integral mentioned in the first part.

I was able to show this indirectly, but I hope that my messy method can be easily surpassed.

Reductive Groups – How to Calculate (Unipotent) Radicals?

My question follows a previous one, essentially this one. I want to understand, given an algebraic group $ G $ (speak linearly) how to calculate its radical and unipotent radical. The (unipotent) residue is the maximal contiguous normally solvable (unipotent) subgroup of $ G $ by definition.

By the Lie-Kolchin theorem, we know that a connected solvable closed subgroup is conjugated to a subset of upper triangular block matrices. Since the radicals are normal subgroups, they consist of upper triangular block matrices.

$$ left (
begin {array} {cccc}
A_1 & star & star & star \
& A_2 & star & star \
& & ddots & star \
& & & A_r
end {array}
right) $$

Then it may be possible to understand on a case by case basis who exactly these subgroups are and to explicitly write down the conditions for belonging to the group $ G $,

But how can one determine the size of the blocks given in Lie-Kolchin's theorem? Typically in the case of $ GL_n $ They are known to be one, and symmetry arguments show that the unipotent radical is trivial. What about other groups, for example? $ GSp (4) $ or a small uniform group?

Equation solution – expression that contains radicals of imaginary numbers

I can not stand a phrase that contains radicals of imaginary numbers,
in the case that it can only be expressed in the form of radicals with real numbers.

For example, I can not stand the expression

Sqrt (2 + i)

there
it can be expressed as

Sqrt (1/2 (2 + Sqrt (5))) + Sqrt (1/2 (-2 + Sqrt (5))) i

But it seems that there is no easy way to do this in Mathematica. I've tried many commands (in Mathematica), but all in vain.

Is there a systematic way to do this work?

Sqrt (2 + i) was a very simple example. I hope the method works for a much more complicated expression.

P.S.
I know that there are many algebraic numbers that can not be expressed as radicals; Root (# ^ 5 + # – 1 &, 1).

Number theory – generalization of a problem in which radicals and the basic function play a role, proposed by Ramanujan in the Journal of the Indian Mathematical Society

The section problem solved from Wikipedia Floor and ceiling functions shows several problems proposed by Ramanujan ((1)). The purpose of this post, if possible, is to generalize some of these identities to positive integers $ n geq 1 $involving fractions or radicals and soil function $ lfloor x rfloor $,

I tried to generalize the identity $ (iii) $, I do not know if my guess identity is in the literature or has good mathematical content. These are my previous failed attempts.

Counterexamples for different formulas.

1) A counterexample of (false) identity
$$ lfloor sqrt (k) {n} + sqrt (k) {n + 1} rfloor = lfloor sqrt (k) {2 ^ k n + k} rfloor $$
is the whole number $ n = $ 525 In the event of $ k = 5 $,

2) counterexamples for the (false) identity
$$ lfloor sqrt (k) {n} + sqrt (k) {n + 1} rfloor = lfloor sqrt (k) {2 ^ k n + 2 (k-1)} rfloor $$
are the integers $ n = 11 $ or $ n = 610 $ In the event of $ k = 6 $,

From this thread of experiments, I get the following guess.

Guess. For every integer $ k geq $ 2 it has the identity
$$ lfloor sqrt (k) {n} + sqrt (k) {n + 1} rfloor = lfloor 2 sqrt (k) {n + frac {1} {2}} rfloor $$
holds over integers $ n geq 1 $,

I do not know if it's easy to prove or if you can find a counterexample.

Question. Do you know whether generalizations (that is, with a good mathematical meaning, with mathematical significance) of the problems suggested by Ramanujan are present in the literature? In this case, please refer to the literature and I try to find and read these generalizations from the literature $ (i) $. $ (ii) $ or $ (iii) $, If this is not in the literature, please add, if possible, a generalization with proof of some of these identities. Especially if you know that my incantation can be proved or disproved by finding a counterexample. Many thanks.

references:

I believe that is the corresponding reference

(1) Srinivasa Ramanujan, Collected papers. Inquiry 723 in p. 332, Providence RI: AMS / Chelsea (2000).