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