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