Zu 3: Bilde ${\displaystyle \sqrt{c}}$

Teil 1

${\displaystyle 38!=2^{e_1}\cdot 3^{e_2}\cdot 5^{e_3}\cdot 7^{e_4}\cdot {11}^{e_5}\cdot {13}^{e_6}\cdot {17}^{e_7}\cdot {19}^{e_8}\cdot {23}^{e_9}\cdot {29}^{e_{10}}\cdot {31}^{e_{11}}\cdot {37}^{e_{12}}}$

${\displaystyle e_1=\lfloor \frac{38}{2}\rfloor +\lfloor \frac{38}{4}\rfloor +\lfloor \frac{38}{8}\rfloor +\lfloor \frac{38}{16}\rfloor +\lfloor \frac{38}{32}\rfloor =19+9+4+2+1=35}$

${\displaystyle e_2=\lfloor \frac{38}{3}\rfloor +\lfloor \frac{38}{9}\rfloor +\lfloor \frac{38}{27}\rfloor =12+4+1=17}$

${\displaystyle e_3=\lfloor \frac{38}{5}\rfloor +\lfloor \frac{38}{25}\rfloor =9}$

${\displaystyle e_4=\lfloor \frac{38}{7}\rfloor =5}$

${\displaystyle e_5=\lfloor \frac{38}{11}\rfloor =3}$

${\displaystyle e_6=\lfloor \frac{38}{13}\rfloor =2}$

${\displaystyle e_7=\lfloor \frac{38}{17}\rfloor =2}$

${\displaystyle e_8=\lfloor \frac{38}{19}\rfloor =2}$

${\displaystyle e_9=\lfloor \frac{38}{23}\rfloor =1}$

${\displaystyle e_{10}=\lfloor \frac{38}{29}\rfloor =1}$

${\displaystyle e_{11}=\lfloor \frac{38}{31}\rfloor =1}$

${\displaystyle e_{12}=\lfloor \frac{38}{37}\rfloor =1}$

${\displaystyle 38!=2^{35}\cdot 3^{17}\cdot 5^9\cdot 7^5\cdot {11}^3\cdot {13}^2\cdot {17}^2\cdot {19}^2\cdot 23\cdot 29\cdot 31\cdot 37}$

Teil 2

${\displaystyle \frac{5039994960}{2}=2519997480}$

${\displaystyle \frac{\dots }{2}=1259998740}$

${\displaystyle \frac{\dots }{2}=629999370}$

${\displaystyle \frac{\dots }{2}=314999685}$

${\displaystyle \frac{\dots }{3}=104999895}$

${\displaystyle \frac{\dots }{3}=34999965}$

${\displaystyle \frac{\dots }{3}=11666655}$

${\displaystyle \frac{\dots }{3}=3888885}$

${\displaystyle \frac{\dots }{3}=1296295}$

${\displaystyle \frac{\dots }{5}=259259}$

${\displaystyle \frac{\dots }{7}=37037}$

${\displaystyle \frac{\dots }{7}=5291}$

${\displaystyle \frac{\dots }{11}=481}$

${\displaystyle \frac{\dots }{13}=37}$

${\displaystyle 5039994960=2^4\cdot 3^5\cdot 5\cdot 7^2\cdot 11\cdot 13\cdot 37}$

Teil 3

${\displaystyle \sqrt{2441348099}\approx 49410}$ ${\displaystyle c=49409\cdot 49411}$