Pair of Large Twin Primes
Theorem
The integers defined as:
- $1\,159\,142\,985 \times 2^{2304} \pm 1$
are a pair of twin primes each with $703$ digits.
Proof
$1\,159\,142\,985 \times 2^{2304} - 1$:
It was checked that it is a prime number using the "Alpertron" Integer factorisation calculator on $6$th March $2022$.
This took approximately $0.2$ seconds.
$1\,159\,142\,985 \times 2^{2304} + 1$:
It was checked that it is a prime number using the "Alpertron" Integer factorisation calculator on $6$th March $2022$.
This took approximately $0.8$ seconds.
$\blacksquare$
Historical Note
David Wells reports in his Curious and Interesting Numbers of $1986$ that this pair of twin primes is the largest reported on by Richard K. Guy in his Unsolved Problems in Number Theory of $1981$.
It is reported that they were discovered by A.O.L. Atkin and N.W. Rickert in $1979$.
Apparently they also discovered the pair $694\,513\,810 \times 2^{2304} \pm 1$ at around the same time.
Sources
- 1979: A.O.L. Atkin and N.W. Rickert: On a larger pair of twin primes: Abstract 79T-A132 (Not. Amer. Math. Soc. Vol. 26: p. A-373)
- 1981: Richard K. Guy: Unsolved Problems in Number Theory
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1159142985 \times 2^{2304} \pm 1$