ფაილების კატალოგი

      Everything of genius is simple.

P_{[10(k-1),10k]}={p: p prime number, 10(k-1)<p<10k}, k=1,2,3...;

P_[1,10n]=U₁ⁿ P_{[10(k-1),10k)]};

п(x)=card{p: p prime number, p<=x};

Theorem. п(10n)<=4n, n=1,2,3,...;                               card P_{[10(k-1),10k]}<=4, k=1,2,3,...

p   r   o   v   e

card P_{[10(k-1),10k]}<=4, k=1,2,3,...;  because of p=1(mod 10) or p=3(mod 10) or p=7(mod 10) or p=9(mod 10), when p>2 and p<> 5.

п(10n)=card P_[1,10n]=card U₁ⁿ P_{[10(k-1),10k]}=SUM₁ⁿ card P_{[10(k-1),10k]}<=SUM₁ⁿ 4=4n. 

                            Emzari Papava