[ May 8, 2009 ] [ Last Update March 4, 2010 ]
Searching for ever larger sporadic palindromes
of the form A^p + B^q (Sum Of Powers)
Patrick De Geest

Let us go fishing palindromes in a vast unlimited pool of numbers.
The rod we use can be for example this formula allowing a freedom
of four variables - two basenumbers A & B and two exponents p & q,
which need not to be consecutive necessarily.

A^p + B^q = palindrome
or with more terms...
A^p + B^q + C^r + ... = palindrome

[ all variables > 1 ]

So thousands or even millions palindromes could be generated.
The choice of these variables is all up to you. Define your own
strategy. Apply the most obscure number theory if you like.
Of course we are mainly interested in the sporadic record solutions
[ sporadic means exclusion of trivial solutions or infinite patterns that might arise ]
that once in a while will pop up on your screen. I am more than happy
to collect and display these record palindromes in the tables below
giving credit to the rightful claimant i.e., its discoverer.
I'll try to maintain a top 100 - or so - of these submissions.

If you spot some beautiful or interesting phenomena/curios but
are not directly record palindromes, report them anyway and I
will publish these as well in a separate section. Palindromic
primes (or palprimes) will be colormarked in their cells resp.

Here are some appetizers !

Here are your contributions !

B.S.Rangaswamy submitted earlier two 63 digit
palindromes comprising of 14 & 13 different constituent powers.
I will present these (and much more material) later in a separate
page dedicated to this devoted contributor.
See the webpage Palindromic Sums of Powers

Aficionado's of Ubasic might like to run my little program code.
[ ps. alas, Ubasic doesn't work with a 64-bit engine like Vista. ]
It is free for use. Amend or improve or optimize it as you like. If
you wrote other/faster code and/or in other languages that you like
to share with my readers please send it in and I will make it public.

   10   color 15:cls
   20   'file SUMPABPQ.UB by Patrick De Geest
   30   Ti="Palindromes of the form A^p+B^q"
   40   print Ti:print
   50   open "sumpower.txt" for append as #1
   60   print #1,"********** ********** ********** **********"
   70   A=2:B=0:Cc=1
   80   input "exponent p ";EP
   90   input "exponent q ";EQ
  100   input "max length palindrome ";MP
  110   loop
  120   X=A^EP+B^EQ
  130   Q=str(X):L=len(Q)
  140   for P=2 to L
  150   if mid(Q,P)<>mid(Q,L+2-P) then cancel for:goto 200
  160   next P
  170   M=str(A)+" ^"+str(EP)+" +"+str(B)+" ^"+str(EQ)+" = "+
        str(X)+" ["+str(alen(X))+" ]"
  180   print #1,M
  190   beep:print Cc;spc(6);A;"^";EP;" +";B;"^";EQ;" = ";:
        color 10:print X;:color 15:print " [";alen(X);"]":inc Cc
  200   inc B
  210   if alen(X)>MP then color 11:print A;"|";B;chr(13);:
        color 15:inc A:B=0
  220   endloop
  230   ' use close #1 after Ctrl+C or Ctrl+Break

Contributors
B.S.Rangaswamy (email)
Jeff Heleen (email)
Jean Claude Rosa (email)