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Math Trek

Primes, Palindromes, and Pyramids

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The number 1888081808881 is interesting on several counts.

First, it's a prime number, evenly divisible only by itself and 1. In base 10, it's also a palindromic number. It has the same sequence of digits whether read forward or backward. In this case, aside from typographical quirks, the number also reads the same upside down or when viewed in a mirror.

In decimal notation, the sequence of palindromic primes begins with 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, and so on.

The largest known palindromic prime, found in 2004 by Harvey Dubner, is 10130036 + 116010611 x 1065014 + 1. It has 130,037 decimal digits.

No one knows yet whether there are infinitely many palindromic primes in base 10.

A palindromic prime pyramid is a sequence of primes in which each term is a palindrome with the previous term as its central digits. So, beginning with the prime 2, you can get the pyramid sequence shown below.

2
9
2
9
3
9
2
9
3
7
3
9
2
9
3
7
3
7
3
9
2
9
3
7
3

Here's another possibility with the same starting point.

2
9
2
9
3
9
2
9
3
3
3
9
2
9
3
3
7
3
3
9
2
9
3
3
7

These two pyramids are the tallest that can be built beginning with the prime 2 and adding one digit at a time to each side (single step).

The idea of studying palindromic prime pyramids was first proposed by G.L. Honaker Jr., a teacher in Bristol, Va. Honaker later collaborated with mathematician and prime specialist Chris K. Caldwell of the University of Tennessee at Martin in a detailed investigation of these curious structures.

It's fairly easy to come up with single-step palindromic prime pyramids starting with the primes 3, 5, and 7. None of these pyramids has more than four levels.

"How can we build them higher?" Honaker and Caldwell ask. Possible strategies include starting with larger primes and allowing the addition of more than one digit to each side with each step.

As it happens, starting with larger primes doesn't help much. You can get a nine-step, truncated pyramid starting with the prime 7159123219517 (found by Felice Russo), but the prospects of doing much better appear limited.

Adding two digits to each side per step, however, is more promising. Starting with 2, the tallest pyramid that can be built has 26 levels. "In fact, there are two pyramids of this height," Caldwell and Honaker remark in a Journal of Recreational Mathematics article on the topic. They found the two examples by doing an exhaustive computer search—building every possible pyramid.

Honaker and Caldwell found that there are three pyramids tied for tallest starting with 3, each of height 28. There's one each starting with the primes 5 and 7, both of height 29. You can find these pyramids and additional information at http://www.utm.edu/staff/caldwell/supplements/.

Based on further computer investigations, Honaker and Caldwell conjecture that all palindromic prime pyramids with fixed step size are finite. They even have a formula that looks promising for estimating the average maximum pyramid height for a given step size.

If the step size is allowed to vary from one row to the next, many more possibilities become available. "For any starting prime we should be able to build as high as we like," Honaker and Caldwell say, "though the taller the pyramids get the larger our step size must be (on average)."

The exception is the prime number 11. In the realm of palindromic primes, 11 is the only one with an even number of digits. Every other palindrome having an even number of digits is divisible by 11 and so can't be prime. So, 11 can't start off a pyramid.

Many questions about palidromic prime pyramids remain open. Is there a better way than exhaustive search for finding the tallest pyramids with fixed step sizes? Can you prove that fixed step size pyramids are finite? Are there efficient strategies for building taller pyramids? What happens in bases other than 10?

There's a lot to ponder in this curious prime realm.

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