# Titanic Prime whose Digits are all 0 or 1

## Theorem

The integer defined as:

- $10^{641} \times \dfrac {10^{640} - 1} 9 + 1$

is a titanic prime all of whose digits are either $0$ or $1$.

That is:

- $\paren 1_{640} \paren 0_{640} 1$

where $\paren a_b$ means $b$ instances of $a$ in a string.

## Proof

It is clear that the digits are instances of $0$ and $1$.

It is also noted that it has $640 \times 2 + 1 = 1281$ 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 $2.5$ seconds.

## Historical Note

According to David Wells in his *Curious and Interesting Numbers, 2nd ed.* of $1997$, this titanic prime was discovered by Harvey Dubner in $1984$, 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): $10^{641} \times \paren {10^{640} - 1} \mathop / 9 + 1$