## nt.number theory – GCD and divisibility of a polynomial

If
$$begin{equation} frac {big((-n^4+27m^4-18m^2n^2)(λ-μ)+8mn^3(-2η+λ+μ)big)} {(3m^2+n^2)^2} end{equation}$$
for integers $$m, n(≠0, ±1)$$ with $$gcd(m, n) = 1$$ is true, (where the numerator is the characteristic polynomial of a 3 by 3 matrix with diagonal entries are $$η neq λneq μ$$).

Then, is there any technique to factorize the numerators of the following equations

$$frac {n^4η+8m^3n(μ-λ)-2m^2n^2big(η-2(λ+μ)big)+m^4big(η+4(λ+μ)big)} {(3m^2+n^2)^2}$$, and
$$frac{-2m(m^2-n^2)Big(n(μ-η)+m(η-2λ+μ)Big)} {(3m^2+n^2)^2}$$ and gives the same numerator as in the first equation ?

Note that in all the above equations, we have $$m, n(≠0, ±1)$$ with $$gcd(m, n) = 1.$$

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## nt.number theory – divisibility of polynomials over partitions

This is a continuation of my earlier MO question.

Given an integer partition $$lambda = ( lambda_1, dots, lambda _ { ell ( lambda)})$$ of $$n$$ Where $$ell ( lambda)$$ is the length of $$lambda$$, associate its conjugate partition $$lambda & # 39;$$. Designate with $$lambda & # 39; & # 39; = lambda & # 39 ;, 0$$ found by adding an extra zero to the right end of $$lambda & # 39;$$. Also define the following two numbers $$a ( lambda & # 39; & # 39;) _ j = lambda_j & # 39; & # 39; – lambda_ {j + 1} & # 39; & # 39;$$ to the $$j = 1,2, dots, ell ( lambda & # 39;)$$ and that too
$$b ( lambda & # 39; & # 39;) = # {j: a ( lambda & # 39; & # 39;) _ j> 0 }$$.

For example when $$lambda = (4,2,1)$$ then $$lambda & # 39; = (3,2,1,1)$$ and $$lambda & # 39; & # 39; = (3,2,1,1,0)$$ and $$a ( lambda & # 39; & # 39;) = (1,1,0,1)$$ and $$b ( lambda & # 39; & # 39;) = 3$$.

Consider the polynomials
$$f_n (q): = sum _ { lambda vdash n} (q-1) ^ {b ( lambda & # 39; & # 39;) – 1} , q ^ { ell ( lambda) -b ( lambda & # 39; & # 39;)}. tag1$$

Designate with $$t_n$$ the biggest $$t$$ so that $$q ^ t$$ Splits $$f_n (q)$$.

QUESTION 1. Is it true that $$t_n in {0,1,2 }$$?

QUESTION 2. (stronger) Is it true that the infinite product $$t_1t_2t_3 cdots = 0 prod_ {k = 1} ^ { infty} 01 ^ {2k} 02 ^ k$$?

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## Divisibility tests in base 12

Can someone give a few divisibility tests in Base 12 with a brief explanation.

I'm not sure how to do it.

Thank you very much.

## python text is printed based on the divisibility of the number

The following code outputs a text based on the divisibility of the number. If the number is not divisible by 3.5 and 7 here, it will be printed out. Does using f string implicitly make the type box from integers to strings?

``````def convert(number):
result = ""
if (number % 3 == 0):
result += "Pling"
if (number % 5 == 0):
result += "Plang"
if (number % 7 == 0):
result += "Plong"
if(number % 7 != 0 and number % 5 != 0 and number%3 != 0):
result = f{'number'}
return result
``````

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## Prove the divisibility test by \$ 7,11,13 for numbers with more than six digits

Prove the divisibility test $$7,11,13$$ for numbers with more than six digits

Attempt:

We know that $$7 cdot 11 cdot 13 = 1001$$, The for a six-digit number, for example, $$120,544$$Let's write it as
$$120544 = 120120 + 424 = 120 cdot1001 + 424$$
So we only check the divisibility of $$424$$ by $$7,11,13$$,

Know a number with more than six digits, for example: $$270060340$$.

$$270060340 = 270270270 – 209930$$
$$= 270 cdot (1001001) – 209930$$
$$= 270 cdot (1001000) + (270 – 209930) = 270 cdot (1001000) – 209660$$

So we check the divisibility of $$209660 = 209209 + 451$$, or only $$451$$,

The test says, however, that: z $$270060340$$we group three digits from the right:
$$270, 60, 340$$
then check the divisibility of $$340 + 270 – (60)$$,

How can you prove that?

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## Divisibility of certain polynomials

Consider the finite sums
$$F_n (q) = sum_ {k = 1} ^ nq ^ { binom {k} 2}$$
with exponents the triangular numbers $$binom {k} 2$$, When $$n$$ strange, it seems that $$F_n (q)$$ does not factor over $$mathbb {Z} (q)$$, On the other hand, if $$n = 2 million$$ is just

QUESTION. is it true that $$F_ {2m} (q)$$ is divisible by the product
$$prod_ {j geq0} (1 + q ^ {m / 2 ^ j})$$
where the product lasts as long as $$m / 2 ^ j$$ is an integer.

Examples. Here is an example:
begin {align} (1 + q ^ 2) (1 + q) , vert & , F_4 (q); qquad (1 + q ^ 3) , , vert , F_6 (q); \ (1 + q ^ 4) (1 + q ^ 2) (1 + q) , , vert & , F_8 (q); qquad (1 + q ^ 6) (1 + q ^ 3) , , vert , F_ {12} (q). end

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## Divisibility when a is odd.

Suppose that $$a | (4b + 5c)$$ and $$a | (2 b + 2 c).$$ Prove that if $$a$$ is so strange $$a | b$$ and $$a | c$$

So, since $$a | (2 b + 2 c)$$ that implies $$a | (4b + 4c)$$ in order to $$a | (4b + 5c)$$ After deducting two dividends, we receive directly $$a | c$$ but I can not understand why it should share $$b$$ if $$a$$ is odd.

## Probability – Divisibility of the square from the sum of two random variables

To let $$I_1, I_2$$ be two independent random variables.
$$I$$ is her sum, $$I = I_1 + I_2$$, Now consider the square of $$I$$.

$$I ^ 2 = (I_1 + I_2) ^ 2$$

We are interested in when $$I ^ 2$$ can be represented as the sum of two independent random variables. In other words, we can find two independent random variables $$X_1, X_2$$ so that
$$I ^ 2 = (I_1 + I_2) ^ 2 = X_1 + X_2$$

Suppose none of $$X_1, X_2$$ are constants.

In general, it's up NOT true.

Question 1: If $$I_1, I_2$$ are infinitely divisible, will be true above?

Question 2: If $$I_1, I_2$$ follows, what kind of special distribution then becomes true above?

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## Infinite divisibility of log normals – MathOverflow

TL; DR: What is the low point of a piece of logarithmic normal distribution?

We know that logarithmic normals are infinitely divisible. What would be the low point of a root of lognormal?

More precisely, let us assume that $$X$$ is a log normal. Given an integer $$k> = 2$$, we now, since there is $$X_1, …, X_k$$ so that:

$$mathcal {L} (X) = mathcal {L} ( sum limits_ {i = 1} ^ {k} X_i)$$

From where $$mathcal {L}$$ denotes the distribution of a random variable.

$$Question:$$ Is there a way to simulate directly from $$X_i$$Law?

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## Divisibility – What makes it so difficult to prove that \$ 3 \$ shares no odd perfect number?

To let $$sigma (x)$$ denote the Sum of divisors the positive integer $$x$$, The frequency Index from $$x$$ is then given by the formula $$I (x) = sigma (x) / x$$,

A number $$N in mathbb {N}$$ it is said that Perfect if $$sigma (N) = 2N$$, Equivalent, if $$N$$ is perfect $$I (N) = 2$$,

Euler showed that one odd perfect number $$M$$, if available, must necessarily have the form $$M = p ^ k m ^ 2$$, from where $$p$$ is the special prime number satisfying $$p equiv k equiv 1 pmod 4$$ and $$gcd (p, m) = 1$$,

Here is my question:

What makes it so difficult to prove that? $$3$$ does not share an odd perfect number?

MY ATTEMPT

So leave it now $$M = p ^ k m ^ 2$$ be an odd perfect number with a special prime $$p$$, Let's assume the opposite $$3 mid$$,

Since $$3 equiv 3 pmod 4$$, then $$p neq 3$$, (Actually, $$p$$ is prime with $$p equiv 1 pmod 4$$ implies that $$p geq 5$$).

So that means that if $$3 mid$$, then $$3 ^ 2 mid n ^ 2 mid$$, Accept that $$3 ^ 2 || M$$ (It means that $$3 ^ 2 mid$$ but $$3 ^ 4 nmid M$$). This implies that $$13 = 1 + 3 + 3 ^ 2 = sigma (3 ^ 2) mid sigma (M) = 2M$$, which means, that $$13 mid$$, This implies that
$$I (3 ^ 2) I (13) leq I (M),$$
provided that $$p = 13$$, or
$$1.60673 approx frac {9507} {5917} = I (3 ^ 2) I ({13} ^ 2) I ({61} ^ 2) I ({97} ^ 2) leq I (M) = 2$$
assumed $${13} ^ 2 || M$$ implies that
$$3 cdot {61} = 183 = 1 + 13 + {13} ^ 2 = sigma ({13 ^ 2}) mid sigma (M) = 2M$$
and
$$3 cdot {13} cdot {97} = 3783 = 1 + 61 + {61} ^ 2 = sigma ({61} ^ 2) mid sigma (M) = 2M,$$
I dare assume that an odd perfect number is not divisible everything the odd primes below $$100$$ (except for $$3$$. $$5$$, and $$7$$ – since we already know that $$105 nmid M$$). However, I have no proof. (Of course, I know that all it takes is a simple calculation.)