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

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?

• 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 $5\cdots 5$ – Peter Apr 29 at 8:30
• 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. – Peter Apr 29 at 8:34