# Titanic Prime whose Digits are all 9 except for one 1

## Theorem

The integer defined as:

- $2 \times 10^{3020} - 1$

is a titanic prime all of whose digits are $9$ except one, which is $1$.

That is:

- $1 \left({9}\right)_{3020}$

where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.

## Proof

It is clear that the digits are instances of $9$ except for the first $1$.

It is also noted that it has $3020 + 1 = 3021$ digits, making it titanic.

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

This took approximately $25.8$ seconds.

## Historical Note

According to David Wells in his *Curious and Interesting Numbers, 2nd ed.* of $1997$, this titanic prime was discovered by Hugh Cowie Williams in $1985$, but this has not been corroborated.

At the time it was the largest such prime number known.

It needs to be investigated whether this record has been broken since then.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2 \times 10^{3020} - 1$