The number \[ 21982145917308330487013369 \] is the thirteenth power of a positive integer. Determine that integer.
Solution
1. One has\[ 89^{13}=21982145917308330487013369. \]
Repeated squaring gives \[ 89^2=7921, \] \[ 89^4=62742241, \] and \[ 89^8=3936588805702081. \] Hence \[ \begin{aligned} 89^{13} &= 89^8\cdot89^4\cdot89\\ &= 21982145917308330487013369. \end{aligned} \]
2. The positive integer in the problem is\[ \boxed{89}. \]
By hypothesis, the displayed number is \(m^{13}\) for some positive integer \(m\). Step 1 shows that it is also \(89^{13}\). The function \[ x\longmapsto x^{13} \] is strictly increasing on the positive real numbers, so \[ m^{13}=89^{13} \] implies \(m=89\).
3. Q.E.D.
Step 2 gives the requested integer.