# Titanic Prime whose Digits are all Prime

## Theorem

The integer defined as:

- $7352 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$

is a titanic prime all of whose digits are themselves prime.

That is:

- $\underbrace{7352}_{275} 7353$

## Proof

It is clear that the digits are instances of $7$, $3$, $5$ and $2$, and so are all prime.

It is also noted that it has $275 \times 4 + 4 = 1104$ 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 $2.1$ 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.

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

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

It was unfortunately misreported by Paulo Ribenboim as $7532 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$, which is composite.

This mistake was propagated by David Wells, who repeated it in his *Curious and Interesting Numbers, 2nd ed.* of $1997$.

## Sources

- 1989: Paulo Ribenboim:
*The Book of Prime Number Records*(2nd ed.) - Sep. 1994: Paulo Ribenboim:
*Prime Number Records*(*College Math. J.***Vol. 25**,*no. 4*: pp. 280 – 290) www.jstor.org/stable/2687612

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $7532 \times \paren {10^{1104} - 1} / \paren {10^4 - 1} + 1$