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.)

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

  • $\begingroup$ 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. $\endgroup$ – Ross Millikan Jul 21 '18 at 20:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.