How many number of non zero pairs $(a,b)$ such that $a,b$ are both palindrome numbers, and the sum of $a$ and $b$ is $A.$.

Given any number let say $$N$$, how many ways this can be written as the sum of the palindrome numbers. For example $$1443$$ there are $$20$$ pairs of palindrome which have sum $$1443$$. $$(1441, 2), (1221, 222), (999, 444), (989, 454), (979, 464), (969, 474), (959, 484), (949, 494), (898, 545), (888, 555), (878, 565), (868, 575), (858, 585), (848, 595), (797, 646), (787, 656), (777, 666), (767, 676), (757, 686) (747, 696)$$

I tried this and able to find all possible pairs for small numbers $$N<=10^{14}$$.

• list all palindromic numbers in $$O(\sqrt N * log N)$$.
• Iterate through the list for all 1....$$\sqrt N$$

    int id = lowerBound(allPanin, n / 2 + 1);

for (int i = 0; i < id; i++) {

int p = Collections.binarySearch(allPanin, n - allPanin.get(i));
if (p > 0){
System.out.println(n - allPanin.get(i) + ", " + allPanin.get(i));
tp += 1;
}
}


But how to find all pairs for $$N <= 10^{18}$$

• This is A260254. The graph looks interesting. Aug 21, 2020 at 16:26