przeto μ = [ ψ ( n ) = ψ ( 316 ) = 65 ] − 14 = 51 ; {\displaystyle \mu =\left[\psi \left({\sqrt {n}}\right)=\psi \left(316\right)=65\right]-14=51;} m + μ = ψ ( n ) = 65 {\displaystyle m+\mu =\psi \left({\sqrt {n}}\right)=65} φ ( n , m ) = φ ( 100 000 , 14 ) = 14204 {\displaystyle \varphi (n,m)=\varphi (100\,000,14)=14204} . Tego trzeba dowieść rachunkiem następującym. Liczby te 14 są następujące: p 1 = 2 , {\displaystyle p_{1}=2,} p 2 = 3 , {\displaystyle p_{2}=3,} p 3 = 5 , {\displaystyle p_{3}=5,} p 4 = 7 , {\displaystyle p_{4}=7,} p 5 = 11 , {\displaystyle p_{5}=11,} p 6 = 13 , {\displaystyle p_{6}=13,} p 7 = 17 , {\displaystyle p_{7}=17,} p 8 = 19 , {\displaystyle p_{8}=19,} p 9 = 23 , {\displaystyle p_{9}=23,} p 10 = 29 , {\displaystyle p_{10}=29,} p 11 = 31 , {\displaystyle p_{11}=31,} p 12 = 37 , {\displaystyle p_{12}=37,} p 13 = 41 , {\displaystyle p_{13}=41,} p 14 = 43 ; {\displaystyle p_{14}=43;}
![{\displaystyle \mu =\left[\psi \left({\sqrt {n}}\right)=\psi \left(316\right)=65\right]-14=51;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b51c22c522e27ebdc4d72f785348d945f5b7077)
. Tego trzeba dowieść rachunkiem następującym.
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2090b2380d877c3c5c5d83e85cee7aa2f2e569e)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d95b07eb3bf7778dd46a1829ace104a1367035)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70576e541966a91c31712a599411cabbe0f517ba)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b09c8a1e96192334c14ac50654d9745512fdcd77)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe83036488684ee0ae70a1f7affb7270db206220)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/327d2941b3b6a2f388a84cd106a585ff69b8c6dc)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef12a4c51aca1018457b85d25a4855fb0a36000)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8863af9633049defc329bb52e67da3adadacf4bd)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bdfadfd9741f6a4d22a4993dc13dd6730dc92d2)
![{\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\,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6dd209d78b044b53ac104e625f6737da874a5f)
