# A Special Observation on Prime Numbers and $\pi (n)$

$$\eth(n)$$ is a little algorithm I made, which may appear to be quite complex, so I will start with an example middle of the post. Questions are at the end of the post.

Definition

Let $$W$$ be the function , defined as $$W(a,b)=r$$

Let's given $$a,b\in \mathbb{N}$$ and $$a>1$$

Take $$m$$ to be the integer s.t. $$a^{m+1} \ge b > a^{m}$$, i.e. $$m = \lceil \log{b}/\log{a} \rceil - 1$$.

Arrange as: $$a^{m+1} - b$$ $$= r_{l} a^l + r_{l-1} a^{l-1}+... + r_1 a^1 + r_0 a^0$$ $$=(r_{l} r_{l-1} ... r_{1} r_{0})_{a}$$

Where $$r=\sum_{i=0}^{l}r_{i}$$

●We can prove easily

$$W(a,b)=r$$ iff $$b+r\equiv 1($$ mod $$a-1)$$

Note $$W(a,0)=$$ not define

Now here $$n \in \mathbb{N}$$

$$S$$ is a function defined as

$$S(a,n)=\sum_{i=1}^{a}i^{n}$$

Let's $$p$$ is prime and $$p+1=z$$

$$\eth$$ is a function defined as

$$\eth (n) = \sum_{W(z,W(z,S(z,2n)))\ne z}1$$

I strongly observed

If $$z>2n+2$$ Then $$W(z,W(z,S(z,2n)))=z$$

Hence I conclude

$$\eth(n)\leq \pi (2n+1)$$

$$\eth(n) \approx \pi (n)$$

$$|\eth(n) - \pi (n)|\leq 2$$

Observation table

$$\begin{array}{c | c | c |c | } n & \eth(n) & \pi(n) \\ \hline 1 & 2 & 0 \\ \hline 2 & 3 & 1 \\ \hline 3 & 3 & 2 \\ \hline 5 &4& 3 \\ \hline 9 &4& 4 \\ \hline 10 &5& 4 \\ \hline 50 &15& 15 \\ \hline 100 &26& 25 \\ \hline 200 &44& 46 \\ \hline \end{array}$$

Example

we want to find $$W(6,W(6,S(6,2)))$$

First calculate $$S(6,2)=1^{2}+2^{2}+...+6^{2}=91$$

$$\implies W(6,W(6,91))$$

Here for calculate $$W(6,91)$$

$$6^{3}-91 = 125 = (325)_{6}$$

$$\implies r = \sum r_{i} = 3+2+5 =10$$

$$hence W(6,91) = 10$$

Again to calculate $$W(6,W(6,91))=W(6,10)$$

$$6^{2}-10 =26 = (42)_{6}$$

$$\implies r=\sum r_{i} = 4+2 =6$$

Hence $$W(6,W(6,S(6,2)))=6$$

Table For $$W(t,W(t,S(t,2)))$$ which helps to calculate $$\eth(1)$$.

$$\begin{array}{c | c | c |c | } t & W(t,S(t,2)) & W(t,W(t,S(t,2))) \\ \hline 2 & 2 & 0 \\ \hline 3^{*} & 3 & 0 \\ \hline 4^{*} & 4 & 0 \\ \hline 5 & 6 & 7 \\ \hline 6^{*} & 10 & 6 \\ \hline 7 &5 & 2 \\ \hline 8^{*} &14& 8 \\ \hline 9 &12& 13 \\ \hline 10 &12& 16 \\ \hline 11 & 15 & 16 \\ \hline 12^{*} & 22 & 12 \\ \hline 13 & 10 & 3 \\ \hline 14^{*} & 26 & 14 \\ \hline 15 & 21 & 22 \\ \hline 16 &20 & 26 \\ \hline 17 &24& 25 \\ \hline 18^{*} &34& 18 \\ \hline 19 &15& 4 \\ \hline 20^{*} &38& 20 \\ \hline 21 &30& 31 \\ \hline \vdots &\vdots & \vdots \\ \hline \end{array}$$

$$t^{*} = z$$

From table $$W(t,W(t,S(t,2)))$$ we can calculate $$\eth(1)$$ by counting $$z$$ such that $$W(z,W(z,S(z,2)))\ne z$$. we can observe it's only happens when $$z=3$$ and $$4$$ hence $$\eth(1)= 2$$

$$\chi$$ is function defined as

$$\chi(n)=\sum_{p \nmid S(p,2n)}1=\sum_{p-1|2n}1$$

And $$n \in \mathbb{N}$$

Proof for $$\chi(n)$$

$$\implies \eth (n)\geq \chi(n)$$

Question

• What is formula for $$\eth(n)$$?

• Can we prove above observation ?

You can check by using below program

'''python

n1= 2
o = 1
while n1 < 300:
m = 2
print("\n n1=",n1)
#print("m=",m)

num=n1
sum_num = 0

for i in range(1, num+1):
sum_num += i**(m)
n2 = (sum_num)
#print("$n1^2m=",n2) rem_array = [] while n2 != 1: mod = n2%n1 if mod != 0: rem = n1-mod n2 = n2 + rem rem_array.append(round(rem)) n2 = n2/n1 else: n2 = n2/n1 rem_array.append(0) #print(rem_array[::-1],sum(rem_array)) #print(sum(rem_array)) n2 = sum(rem_array) rem_array = [] while n2 != 1: mod = n2%n1 if mod != 0: rem = n1-mod n2 = n2 + rem rem_array.append(round(rem)) n2 = n2/n1 else: n2 = n2/n1 rem_array.append(0) #print(rem_array) #print(rem_array[::-1],sum(rem_array)) if(n1 == sum(rem_array)): print("W(",n1,",W(",n1,",S(",n1,",m)))=",n1) #else: #print("not ok") n1 += o  ''' Note ● if $$W(a,(a-1)q)=a$$ Then $$q\in$$ {$$a^{n}+(-1)^ni | i=$${$$1,2,..,(a+n-2)$$}$$\forall n \in \mathbb{W}$$} *I think ,$$\eth$$ function very much impact on prime number theorem, thank you. •$\Lambda$and$\Omega\$ both have standard meanings in Analytic Number Theory, so they are not good choices for names of your functions. – Gerry Myerson Jun 26 at 2:54
• @Gerry_Myerson Yes you are right now it's changed – Pruthviraj Jul 12 at 14:12
• Take a look at the other questions to see how you need to improve yours (simplify, shorter, motivated) – reuns Jul 12 at 21:10