Problems & Puzzles: Puzzles
Puzzle 14.- Pal-Primes and sum of powers
Lets define the following kind of "Palprimes" (palindromic primes) : those that are the sum of the same power of consecutive numbers
Ppal = x n +(x+1) n
Jud. McCranie sent us a mail (Fri. 17 Jul. 1998)
For the part b) of this puzzle he wrote :
"Pal-primes and sum of powers - why not any primes for odd powers?
Because if n is odd then x^n + (x+1)^n is divisible by 2x+1. If n > 1 then 2x+1 is a proper divisor of x^n+(x+1)^n, so it isn't prime. There are, of course, solutions when n=1".
Patrick De Geest also wrote us (Sat, 18 Jul 1998) for the same subject :
"I visited puzzle page 014 (Palprimes and sum of powers). Did you know I started a search palindromic sums of squares of two consecutive numbers ? The largest one I found ( unfortunately not a prime)is the following 17 digit number : 80.472.264^2 + 80.472.265^2 = 12951570707515921. The next three ones (181,313,3187813) are palindromic primes :
9^2 + 10^2 = 181
So, only two more to find, but it will be difficult ! I didn't check for other powers. Two years of delving into palindromic numbers revealed to me that palindromes in general don't thrive in higher dimensions, so I think that solutions in higher than 2 powers will be very, very hard to find".