Prime Palindromes with Even Number of Digits

Is There Another Prime Number Besides Eleven That Is Also a Palindrome and Has an Even Number of Digits?

Introduction to Palindromic Primes

The question of whether there exists another prime number that is also a palindrome with an even number of digits is an intriguing one. To answer this, let's dive into the characteristics and properties of such numbers. Palindromic primes are prime numbers that remain the same when their digits are reversed. Famous examples include 11, which is the only two-digit palindromic prime. Another example is 101, which is a palindrome but has an odd number of digits and is thus not part of our objective.

Understanding Palindromic Primes with Even Digits

Upon inspection, the next possible candidate after 101 is 1001, but it is not prime. Indeed, 1001 can be divided by 7 and 143. It quickly becomes evident that no other palindromic prime with an even number of digits exists beyond 11. The answer is a definitive No.

Divisibility Test and Proof

The key to understanding why there are no other even-length palindromic primes lies in the divisibility test for 11 and an interesting characteristic observed by Jovan Radenkovic.

Divisibility by 11

To determine if a number is divisible by 11, we use the test where we subtract the sum of the digits in the odd positions from the sum of the digits in the even positions. If the result is 0 or divisible by 11, then the number is divisible by 11. However, for even-length palindromic numbers, this test reveals a crucial property: every pair of digits in the middle is canceled out, making the number divisible by 11.

Mathematical Proof by Jovan Radenkovic

Jovan Radenkovic's observation highlights that in an even-length palindrome, the number is divisible by 11. Let's explore the underlying mathematics. Consider a number in base b with an even length expressed as d_1d_2…d_{n-1}d_nd_nd_{n-1}…d_2d_1. When expanded, the number can be expressed as a sum of terms of the form d_ib^i. Modulo b-1, b is -1, leading to an alternating sum where all terms cancel out. This alternating sum results in a remainder of 0 when divided by b-1, proving that the number is indeed divisible by 11.

Conclusion

In summary, through the examination of palindromic primes and the properties of even-length palindromic numbers, we conclude that the only prime number that is also a palindrome and has an even number of digits is 11. No other such prime exists. This unique property of 11 not only highlights its special status among prime numbers but also opens up an interesting area of study in number theory.