Are there natural number satisfying the following equations?

The motivation comes from the following question on MathOverflow:

Is the exponential version of Catalan-Dickson conjecture true?

Question 1. Is there a natural number $n$ satisfying the equation $n(n+1)=2^{[\log^{n}_{2}]+2}$ where $[.]$ indicates the floor function. I am also interested in the minimum $n$ satisfying this equation if there is any.

As the answer to the above question is negative, let's consider the following more generalized form:

Question 2. Are there natural number $n$ and prime number $p$ satisfying the equation $\frac{n(n+1)}{2}=\frac{p}{p-1}\times p^{[\log_{p}^{n}]}+\frac{p-2}{p-1}\times p^n$ where $[.]$ indicates the floor function. (Note that for $p=2$ we get the previous equation.)

• Well for $n>1$ the left hand side would not be a power of $2$, so no. – Gal Porat Jul 21 '18 at 9:33
• Anyway, what is $\log_2^n$? – Hagen von Eitzen Jul 21 '18 at 9:35
• For the more general question, $\frac{p-2}{p-1}\cdot p^n$ quickly outgrows $\frac{n(n+1)}{2}$... – Wojowu Jul 21 '18 at 10:43

The RHS of $$n(n+1)=2^{[\log^{n}_{2}]+2}$$ is a power of $2$ while the LHS is not a power of $2$ unless $n=1$ and $n=1$ is not a solution.

Thus I would say there are no solutions for this equation.

• And for the second question the left side is an integer while the right side is probably not. It is difficult to get rid of the $p-1$ in the denominator. – Ross Millikan Jul 21 '18 at 20:51