Pair of Titanic Twin Primes

Theorem

The integers defined as:

$190 \, 116 \times 3003 \times 10^{5120} \pm 1$

are a pair of titanic twin primes.


That is:

$570 \, 918 \, 347 \paren 9_{5820}$

and:

$570 \, 918 \, 348 \paren 0_{5819} 1$


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


Proof

It is noted that these integers have $9 + 5820 = 5829$ digits, making them titanic.

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

This took approximately $45$ seconds.


Historical Note

According to David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$, this pair of titanic twin primes was discovered by Harvey Dubner on $5$ October $1995$, but this has not been corroborated.


Sources

  • Feb. 1996: Harvey Dubner: Numbers Count (Personal Computer World )
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $190,116 \times 3003 \times 10^{5120} - 1$ and $190,116 \times 3003 \times 10^{5120} + 1$
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