On finding all positive integers $a,b$ such that $b\pm a$ and $ab$ are palindromic

Abstract

It is proven that the only integer solutions $(a,b)$ such that $a+b$ and $ab$ are palindromic are $(2,5\cdot 10^k-3)$, $(3,24)$ and $(9,9)$, and in a similar fashion, $b-a$ and $ab$ are only palindromic at $(a,b)=(3,147\cdot 10^{4(k+1)}+5247\sum_{i=0}^k10^{4i})$, $(3,161\,247\cdot 10^{4k+7}+5247\sum_{i=0}^k10^{4i+3}+387)$, $(3,147)$ and $(3,161\,247\,387)$ for $k=0,1,2,\cdots$. Note $a\le b$ without loss of generality.