Reading the question can the product of four positive integers in A.P. be a square?, also made me question whether the product of $n$ positive integers, where $n \gt 5$ in arithmetic progression be a palindrome?

Me and user Peter tried to find solutions for various $n$ in PARI/GP, and found that there were no solutions for a large range of numbers we tested when $n \gt 5$.

This led to the following questions:

($1$) Can the product of $n$ positive integers, where $n \gt 5$ in A.P. be a palindrome?

($2$) If yes then for what value of $n$ is there no solution?

  • 1
    $\begingroup$ There are slight restrictions that accelerate the search : The difference $d$ cannot be coprime to $10$ , and if it is congruent to $2,4,6$ or $8$ modulo $10$, the only possible palindrome is $5\cdots 5$ $\endgroup$ – Peter Apr 29 at 8:30
  • 1
    $\begingroup$ A general lower bound for the value of the product is $$(n-1)!\cdot 2^{n-1}$$ which makes it unlikely, that a solution for large $n$ exists. $\endgroup$ – Peter Apr 29 at 8:34

Your Answer

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

Browse other questions tagged or ask your own question.