Problems & Puzzles: Puzzles

Let’s 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

1. Can you find the first 5 of them for each even power ?
2. ( a minor question) why there are not any primes for odd powers ?

Solution

Jud. McCranie sent us a mail (Fri. 17 Jul. 1998)

"For n=2: 

x	Ppal
1	5
9	181
12	313
1262	3187813

 and the 5th one has eluded me. for n=4: x=9, Ppal = 16561

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".

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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
12^2 + 13^2 = 313
1262^2 + 1263^2 = 3187813

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".

***On April 24, 2026 Patrick De Geest wrote

I have finally found the fifth prime of the form n^2 + (n+1)^2

Besides
n = 1            Ppal = 5
n = 9            Ppal = 181
n = 12           Ppal = 313
n = 1262         Ppal = 3187813

we can now add...beware, it is a whopper...

n = 91732095351342012927350087594  [29]
and
Ppal = 16829554635095405876254045429592454045267850459053645592861  [59]

Note also that the basenumber and the palindrome both have a prime length (29 & 59).

If you like you might add a link to my webpage where I keep track of all findings large and small https://www.worldofnumbers.com/sumsquare.htm

BTW, I used a code made by Robert Xiao. It is a very fast program written for finding palindromic quadratic solutions using the NVIDIA gpu of my graphics card.

More info at https://www.worldofnumbers.com/distributed.htm

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