# Smallest Titanic Palindromic Prime

## Theorem

The smallest titanic prime that is also palindromic is:

- $10^{1000} + 81 \, 918 \times 10^{498} + 1$

which can be written as:

- $1 \underbrace {000 \ldots 000}_{497} 81918 \underbrace {000 \ldots 000}_{497} 1$

## Proof

It was checked that it is a prime number using the "Alpertron" Integer factorisation calculator on $6$th March $2022$.

This took approximately $1.7$ seconds.

It remains to be demonstrated that it is the smallest such palindromic prime with $1000$ digits or more.

By 11 is Only Palindromic Prime with Even Number of Digits, there are no palindromic primes with exactly $1000$ digits.

Hence such a prime must be greater than $10^{1000}$.

We need to check all numbers of the form:

- $1 \underbrace {000 \ldots 000}_{497} abcba \underbrace {000 \ldots 000}_{497} 1$

with $\sqbrk {abc} < 819$.

Using the Alpertron integer factorization calculator and the argument:

x=0;x=x+1;x<820;10^1000+x*10^500+RevDigits(x/10+10^499,10)

it is verified that there are no primes in the range $\sqbrk {abc} < 819$.

Therefore the number above is the smallest titanic palindromic prime.

$\blacksquare$

## Sources

- 1994: Harvey Dubner:
*Palindromic Primes with a Palindromic Prime Number of Digits*(*J. Recr. Math.***Vol. 26**,*no. 4*: p. 256)

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $10^{1000} + 81,918 \times 10^{498} + 1$