Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation $$p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$$ with the restrictions $x<2n$ and $x_i\le n$ for all $i$.

Observe that for the numbers $x_0=\dots=x_{p^n}=p^n$ and $x=2n$ we have $$\sum_{i=0}^{p^n}p^n=(p^n+1)\cdot p^n=p^{x}+p^n$$so, the restriction $x<2n$ is necessary for ruling out this obvious solution.

Question 1. Determine $p$ and $n$ for which the Diophantine exponential equation $p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$ does not have positive integer solutions $x,x_0,\dots,x_{p^n}$ with $x<2n$ and $x_i\le n$?

Question 1 can be also reformulated as follows.

Question 2. Determine $p$ and $n$ for which the Diophantine exponential equation $p^x+p^n=\sum_{k=1}^n x_kp^k$ does not have non-negative integer solutions $x,x_0,\dots,x_n$ with restrictions $n<x<2n$ and $\sum_{k=1}^nx_k=2^n+1$?

Question 3. What is the answer to Questions 1,2 for $p=2$?

Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation $$p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$$ with the restrictions $x_i<n$ for all $i$.

Observe that for the numbers $x_0=\dots=x_{p^n}=n$ and $x=2n$ we have $$\sum_{i=0}^{p^n}p^{x_i}=(p^n+1)\cdot p^n=p^{x}+p^n$$so, the restriction $x_i<n$ is imposed for ruling out this obvious solution.

Question 1. Determine $p$ and $n$ for which the Diophantine exponential equation $p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$ does not have positive integer solutions $x,x_0,\dots,x_{p^n}$ with $x_i<n$?

Question 1 can be also reformulated as follows.

Question 2. Determine $p$ and $n$ for which there exists a polynomial $f(x)$ of degree $<n$ with non-negative integer coefficients such that $f(0)=0$, $f(1)=p^n+1$ and $f(p)=p^n+p^x$ for some $x>n$.

Question 3. What is the answer to Questions 1,2 for $p=2$?

Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation $$p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$$ with the restrictions $x<2n$ and $x_i\le n$ for all $i$.

Observe that for the numbers $x_0=\dots=x_{p^n}=p^n$ and $x=2n$ we have $$\sum_{i=0}^{p^n}p^n=(p^n+1)\cdot p^n=p^{x}+p^n$$so, the restriction $x<2n$ is necessary for ruling out this obvious solution.

Question 1. Determine $p$ and $n$ for which the Diophantine exponential equation $p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$ does not have positive integer solutions $x,x_0,\dots,x_{p^n}$ with $x<2n$ and $x_i\le n$?

Question 2. What is the answer to Question 1 for $p=2$?

Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation $$p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$$ with the restrictions $x<2n$ and $x_i\le n$ for all $i$.

Observe that for the numbers $x_0=\dots=x_{p^n}=p^n$ and $x=2n$ we have $$\sum_{i=0}^{p^n}p^n=(p^n+1)\cdot p^n=p^{x}+p^n$$so, the restriction $x<2n$ is necessary for ruling out this obvious solution.

Question 1. Determine $p$ and $n$ for which the Diophantine exponential equation $p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$ does not have positive integer solutions $x,x_0,\dots,x_{p^n}$ with $x<2n$ and $x_i\le n$?

Question 1 can be also reformulated as follows.

Question 2. Determine $p$ and $n$ for which the Diophantine exponential equation $p^x+p^n=\sum_{k=1}^n x_kp^k$ does not have non-negative integer solutions $x,x_0,\dots,x_n$ with restrictions $n<x<2n$ and $\sum_{k=1}^nx_k=2^n+1$?

Question 3. What is the answer to Questions 1,2 for $p=2$?