# Titanic Prime whose Digits are all Odd

## Theorem

The integer defined as:

- $1358 \times 10^{3821} - 1$

is a titanic prime all of whose digits are odd.

That is:

- $1357 \paren 9_{3821}$

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

## Proof

It is clear that the digits are all instances of $9$ except for the initial $1357$, all of which are odd.

It is also noted that it has $4 + 3821 = 3825$ 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 $34.4$ 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 $1988$, but this has not been corroborated.

He also reports that it is the largest known prime whose digits are odd.

This was likely the case in $1997$, but research is needed to see if this record has since been superseded.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1358 \times 10^{3821} - 1$