Intereting Posts

Determinant value of a square matrix whose each entry is the g.c.d. of row and column position
boundary of the boundary of a set is empty
intuition behind weak solution
$\left \| \cdot \right \|$ is an induced norm. If $\left \| A \right \|<1$, how to show that $I-A$ is nonsingular and …?
Result and proof on the conditional expectation of the product of two random variables
Formulae of the Year $2016$
Find the limit $\lim \limits_{n\to \infty }\cos \left(\pi\sqrt{n^{2}-n} \right)$
Composition of two reflections (non-parallel lines) is a rotation
Maximum of $|(z-a_1)\cdots(z-a_n)|$ on the unit circle
Single variable complex analysis vs the world of the functions $f:\Bbb R^2 \to \Bbb R^2$.
Products of homology groups
What can we learn from prime generating polynomials?
Does a non-trivial solution exist for $f'(x)=f(f(x))$?
Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$
Is a polynomial equation of degree $\ge 5$ not solvable by any way?

**Description:** We can write, $n!= 2^s \times a \times b \cdots (1)$

where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$ .**It is given that,** $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$.

**Claim:** If $n!= 2^s \times a \times b $ and $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$, then $a$ has $90 \% $ primes less than $n$ as factor.

- How many ways $12$ persons may be divided into three groups of $4$ persons each?
- number of subsets of even and odd
- Binomial Identity $\sum\binom{2n+1}{2k+1}\binom{m+k}{2n} = \binom{2m}{2n}$
- Prove that there are two frogs in one square.
- Number of ways to partition a rectangle into n sub-rectangles
- Calculating the total number of surjective functions

**Proof:**

Here, $\nu_ p(n)$ denotes the p-adic valuation of $n$ and $s_p(n)$ denotes the sum of the standard base-p digits of $n$, so,

$\nu _{p}(n!)={\frac {n-s_{p}(n)}{p-1}}$ ($\textit{Legendre’s formula}$). For $p = 2 $, we obtain $\nu _{2}(n!)= n-s_{2}(n)$, where $s_{2}(n) $ is the number of $1$’s in the binary representation of $n$. The number of primes less than $n$ is denoted by $\pi(n)$. By $\textit{Prime Number Theorem}$, $\pi(n)\approx \frac{n}{\log n}$.

Since, $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2} $

$\implies \log_2(a )-\log_2(b)= \log_2 (2)^{s-2}$ [ignoring a quantity $<1$ on R.H.S]

$ \implies \log_2(a )-\log_2(b )=n- s_2(n)-2$

$ \implies \sum_{p \mid a}\log_2( p ^{\nu _{p}(n!)})-\sum_{q \mid b}\log_2( q ^{\nu _{q}(n!)})=n- s_2(n)-2 \dots (2)$

Equation $(2)$ implies that,

$\sum_{p \mid a}\log_2( p ^{\nu _{p}(n!)})-\sum_{q \mid b}\log_2( q ^{\nu _{q}(n!)}) >0 \cdots (3)$

Now, we consider the smallest prime factor, $p_0$ of $a$ and the biggest prime factor$q_0$ of $b$. Note, $a$ has $i$ distinct prime factors, $b$ has $j$ prime factors and $ i+j = \pi(n)-1\approx \frac{n}{\log n}-1$. Since,

$\sum_{p \mid a}\log_2( p ^{\nu _{p}(n!)}) < (\log_2(p_0 ^{\nu _{p_0}(n!)}) ) \times i$

and,

$\sum_{q \mid b}\log_2( q ^{\nu _{q}(n!)}) >(\log_2( q_0^{\nu _{q_0}(n!)}) ) \times j$ (See 1, Theorem 3.9 on page 7), so-

$(\log_2( p_0 ^{\nu _{p_0}(n!)}) ) \times i-(\log_2(q_0^{\nu _{q_0}(n!)}) ) \times j>\sum_{p \mid a}\log_2( p ^{\nu _{p}(n!)})-\sum_{q \mid b}\log_2(q ^{\nu _{q}(n!)}) \cdots (4) $

$\implies (\log_2(p_0 ^{\nu _{p_0}(n!)}) ) \times i-(\log_2( q_0 ^{\nu _{q_0}(n!)}) ) \times j>0$

$\implies i \times \log_2( p_0 ^{\nu _{p_0}(n!)}) > j \times \log_2( q_0 ^{\nu _{q_0}(n!)} )$

$\implies i > j \times \frac{ \log_2( q_0 ^{\nu _{q_0}(n!)}) }{\log_2( p_0 ^{\nu _{p_0}(n!)}) }$

Let, $T=\frac{ \log_2( q_0 ^{\nu _{q_0}(n!)}) }{\log_2(p_0 ^{\nu _{p_0}(n!)}) }$.

$\therefore i > (\pi(n)-1-i) \times T$

$\implies i > T \times (\pi(n) -1)- T \times i \implies i> \frac{T}{(1+T)} \times (\pi(n) -1)\cdots (5)$

If $ \frac{T}{(1+T)} \approx 0.9$ so, $i> 0.9 \times (\pi(n)-1)$.

**Query:** Is the above proof correct? Please, let me know if anything is inconsistent.

**Reference :**

- Diophantine equations involving arithmetic functions of factorials, Daniel M. Baczkowski, A Thesis for Master’s of Arts, Miami University.

- A game of guessing a chosen number
- The $\gcd$ operator commutes with functions defined by linear recurrence relations
- Probability of getting A to K on single scan of shuffled deck
- Polynomial $p(a) = 1$, why does it have at most 2 integer roots?
- Knight returning to corner on chessboard — average number of steps
- Number of ways to put N indistinct objects into M indistinct boxes
- Prove that every positive integer $n$ has a unique expression of the form: $2^{r}m$ where $r\ge 0$ and $m$ is an odd positive integer
- Prove that the equation $x^2-y^2 = 2002$ has no integer solution
- Minimum number of points chosen from an $N$ by $N$ grid to guarantee a rectangle?
- Comparing $2013!$ and $1007^{2013}$

Since, $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2} $

$\implies \log_2(\lfloor a \rfloor)-\log_2(\lfloor b \rfloor)= \log_2 (2)^{s-2}$…

This is certainly not correct. What happens if (for example) $a=13,b=3$?

- Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.
- How to tell whether a scheme is reduced from its functor of points?
- Over $\mathbb{R}$, if $Z(p') \subset Z(p)$ when does $p' \vert p$?
- How to prove following integral equality?
- Conjugacy classes in group extensions
- checking if a 2-form is exact
- Rudin: Problem Chp3.11 and need advice.
- Logic Puzzle of the age of three sons
- How to apply Gaussian kernel to smooth density of points on 2D (algorithmically)
- Evaluate $\int_0^\pi\frac{1}{1+(\tan x)^\sqrt2}\ dx$
- Conditional probabilities from a joint density function
- Simplified form for $\frac{\operatorname d^n}{\operatorname dx^n}\left(\frac{x}{e^x-1}\right)$?
- Free groups: unique up to unique isomorphism
- A ring isomorphic to its finite polynomial rings but not to its infinite one.
- Characteristic function of a standard normal random variable