# Strona:A. Baranowski - O wzorach.pdf/8

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przeto ${\displaystyle \mu =\left[\psi \left({\sqrt {n}}\right)=\psi \left(316\right)=65\right]-14=51;}$ ${\displaystyle m+\mu =\psi \left({\sqrt {n}}\right)=65}$
${\displaystyle \varphi (n,m)=\varphi (100\,000,14)=14204}$. Tego trzeba dowieść rachunkiem następującym.
Liczby te 14 są następujące: ${\displaystyle p_{1}=2,}$ ${\displaystyle p_{2}=3,}$ ${\displaystyle p_{3}=5,}$ ${\displaystyle p_{4}=7,}$ ${\displaystyle p_{5}=11,}$ ${\displaystyle p_{6}=13,}$ ${\displaystyle p_{7}=17,}$ ${\displaystyle p_{8}=19,}$ ${\displaystyle p_{9}=23,}$ ${\displaystyle p_{10}=29,}$ ${\displaystyle p_{11}=31,}$ ${\displaystyle p_{12}=37,}$ ${\displaystyle p_{13}=41,}$ ${\displaystyle p_{14}=43;}$

${\displaystyle \varphi (100\,000,14)=\varphi (100\,000,13)-\left[\varphi \left({\frac {100\,000}{43}},13\right)=\varphi \left(\,\,\,\,2325,13\right)\right]=14\,540-\,\,\,\,\,\,\,336=14\,204\,\,}$
${\displaystyle \varphi (100\,000,13)=\varphi (100\,000,12)-\left[\varphi \left({\frac {100\,000}{41}},12\right)=\varphi \left(\,\,\,\,2439,12\right)\right]=14\,899-\,\,\,\,\,\,\,359=14\,540\,\,}$
${\displaystyle \varphi (100\,000,12)=\varphi (100\,000,11)-\left[\varphi \left({\frac {100\,000}{37}},11\right)=\varphi \left(\,\,\,\,2707,11\right)\right]=15\,305-\,\,\,\,\,\,\,406=14\,899\,\,}$
${\displaystyle \varphi (100\,000,11)=\varphi (100\,000,10)-\left[\varphi \left({\frac {100\,000}{31}},10\right)=\varphi \left(\,\,\,\,3225,10\right)\right]=15\,805-\,\,\,\,\,\,\,500=15\,305\,\,}$
${\displaystyle \varphi (100\,000,10)=\varphi (100\,000,\,\,\,9)-\left[\varphi \left({\frac {100\,000}{29}},\,\,\,9\right)=\varphi \left(\,\,\,\,3448,\,\,\,9\right)\right]=16\,361-\,\,\,\,\,\,\,556=15\,805\,\,}$
${\displaystyle \varphi (100\,000,\,\,\,9)=\varphi (100\,000,\,\,\,8)-\left[\varphi \left({\frac {100\,000}{23}},\,\,\,8\right)=\varphi \left(\,\,\,\,4347,\,\,\,8\right)\right]=17\,103-\,\,\,\,\,\,\,742=16\,361\,\,}$
${\displaystyle \varphi (100\,000,\,\,\,8)=\varphi (100\,000,\,\,\,7)-\left[\varphi \left({\frac {100\,000}{19}},\,\,\,7\right)=\varphi \left(\,\,\,\,5263,\,\,\,7\right)\right]=18\,053-\,\,\,\,\,\,\,950=17\,103\,\,}$
${\displaystyle \varphi (100\,000,\,\,\,7)=\varphi (100\,000,\,\,\,6)-\left[\varphi \left({\frac {100\,000}{17}},\,\,\,6\right)=\varphi \left(\,\,\,\,5882,\,\,\,6\right)\right]=19\,181-\,\,\,\,1128=18\,053\,\,}$
${\displaystyle \varphi (100\,000,\,\,\,6)=\varphi (100\,000,\,\,\,5)-\left[\varphi \left({\frac {100\,000}{13}},\,\,\,5\right)=\varphi \left(\,\,\,\,7692,\,\,\,5\right)\right]=20\,779-\,\,\,\,1598=19\,181\,\,}$
${\displaystyle \varphi (100\,000,\,\,\,5)=\varphi (100\,000,\,\,\,4)-\left[\varphi \left({\frac {100\,000}{11}},\,\,\,4\right)=\varphi \left(\,\,\,\,9090,\,\,\,4\right)\right]=22\,857-\,\,\,\,2078=20\,779\,\,}$