Definitions and restrictions

Rare   numbers are positive integers    n    where:

    n    is expressed in base ten
    r    is the reverse of    n      (decimal digits)
    n    must be non-palindromic   (n ≠ r)
    (n+r)    is the   sum
    (n-r)    is the   difference   and must be positive
   the   sum   and the   difference   must be perfect squares


Task

   find and show the first   5   rare   numbers
   find and show the first   8   rare   numbers       (optional)
   find and show more   rare   numbers                (stretch goal)


Show all output here, on this page.



References

   an   OEIS   entry:   A035519          rare numbers.
   an   OEIS   entry:   A059755   odd rare numbers.
   planetmath entry:   rare numbers.     (some hints)
   author's  website:   rare numbers   by Shyam Sunder Gupta.     (lots of hints and some observations).


rare     =:  ( np@:] *. (nbrPs rr) ) b10
  np     =:  -.@:(-: |.)    NB. Not palindromic  
  nbrPs  =:  > *. sdPs      NB. n is Bigger than R and the perfect square constraint is satisfied
    sdPs =:  + *.&:ps -     NB. n > rr and both their sum and difference are perfect squares  
    ps   =:  0 = 1 | %:     NB. Perfect square (integral sqrt)
  rr     =:  10&#.@:|.      NB. Do note we do reverse the digits twice (once here, once in np)
  b10    =:  10&#.^:_1      NB. Base 10 digits