The idea is to choose m and n (and r) and consider the arithmetic progression (10^n+1)(pt+j) + (10^m +10^r)(qt+k). Although it won't be a palindrome for some choices of m and r and n, there is a lot of room to play in, and perhaps you can find ranges of t that guide your choice of m,n and r. If so, you have reduced two progressions to one, hopefully making it easier to study and predict. Gerhard "Maybe Consider Several Choices Simultaneously" Paseman, 2015.05.07

You might look at finding an expression for f(f(n)) and f(f(f(n))), just to see how things might bounce around. In particular, finding n and r so that floor nr preserves parity would be a useful tool for this problem. Gerhard "Might Also Make Pretty Pictures" Paseman, 2015.05.06

Consider combining the systems into one. For n large and t small, pt + j is a palindrome implies (10^n + 1)(pt+j) is a palindrome. Consider how small t can be for both expressions to be palindromic, then pick n and m very large and with the right parity and consider pt+j ... qt+k ... qt+k ... pt+j. You can write this as a single recurrence that hopefully will produce a palindrome for some t depending on m and n. Gerhard "Give Yourself Plenty Of Room" Paseman, 2015.05.06

I see the example with both a and b greater than -1/2. I guess that will be the problematic section. Gerhard "Time To Get A Pencil" Paseman, 2015.05.05

@KurisutoAsutora, my intent is to support the claim that the infimum is not far from exp(-ck log k), instead of the exp (-ck) you suggest might occur. I doubt the data supports either conclusion, but the fact is that the infimum does not occur as a ratio of consecutive smooth numbers for some small k, and this is likely to make a lower bound proof harder. Even if you focus on those cases for which the infimum does occur at consecutive smooth numbers, I don't know how many there are of such cases. Gerhard "Sorry For The Machine Error" Paseman, 2015.05.04

Returning to the millions range, my program reports further improvements, one of them being 221669902,1 for $y,b$. Gerhard "Paging Experts On Stormer's Theorem" Paseman, 2015.04.28

@Arupinski: rename your MathOverflow bookmark to point to Joseph's user page. Gerhard "Saves A Lot Of Time" Paseman, 2015.04.19

It would seem that what is known falls short of $p_{n+1} - p_n \gt \log p_n(\log(\log p_n))^k$ being true for fixed $k \geq 1$ and infinitely many $n$. (I think that) Joseph and I would appreciate if you can confirm/deny this assertion, especially the $k \geq 1$ part. Gerhard "Lost In Number Theory Lumberyard" Paseman, 2015.04.19

After some minimal checking, I got some of the bases wrong. As far as I know, not even logp_n(loglog p_n)^k is known to occur for infinitely many n and $k > 1$. I will update when I get the numbers straightened out. Maier, Pomerance, Pintz, Tao, Green, Ford, Kolyvagin, and Maynard are still some of the names to check. Gerhard "Or Use A Phone Book" Paseman, 2015.04.18

You have results of Maier and Pomerance which say there are (on average maybe?) infinitely many for some real values of $k$ larger than 1. My current investigations and various conjectures suggest your question has the answer yes only for $k$ less than 2. As a start, try Helmut Maier's Chains of large gaps between consecutive primes, done in 1981. Terry Tao announced joint work with four other authors on large gaps, available on ArXiv 1412, with (I think) an upcoming improvement on Maier's result in a followup article. Gerhard "Hope I Got Bases Right" Paseman, 2015.04.18

Thanks for your response. Also, to make the notification system work, use @Gerhard instead of @ Gerhard (@Will in my case). Gerhard "Better Than Firing At Brian" Paseman, 2015.04.17

Wow! In spite of my comments above, I never thought of investigating the lattice of topologies of X itself. Is there any reference investigating the notion of equivalence relations on the lattice of topologies which pertain to being bijectively related? More importantly (for my universal algebraic background) are there important lattice congruences (equivalence relations preserving finite meets and joins) that one could use to help investigate this? Gerhard "Lattices Turn My Mind Inside-Out" Paseman, 2015.04.17

It convinces me that a maximal solution will be a parallelogram with no sides parallel to the original, and that it is a matter of computing the area when the line through R induces such a parallelogram. This area should vary as a simple quadratic in the x-intercept of the line through R, whose maximum is easily determined. Gerhard "Still Working On The Details" Paseman, 2015.04.17