Twelfth Mersenne Prime

Mersenne's twelfth prime is a natural number $2^{127}-1=170141183460469231731687303715884105727$ ${\ displaystyle 2 ^ {127} -1 = 170141183460469231731687303715884105727}$ ${\ displaystyle 2 ^ {127} -1 = 170141183460469231731687303715884105727}$ . It was the largest known prime number from 1876 to 1951.

Content

In Math

This number is the twelfth prime number among the Mersenne numbers ^[1] . This means that there is no number less than this that would have a period of 127 in the binary system when accessed. Eduard Luke showed in the year 1876 that this number is prime with the help of the Lucas-Lemer simplicity test . This number remained the largest known prime number for 75 years, until 1951, when it was shown that ( ^2,148 + 1) / 17 is an even larger prime number. This number is also the fourth double Mersenne number and the fifth Catalan - Mersenne number (the largest known prime in both cases). Verification of the simplicity of the next Catalan - Mersenne number by known methods is impossible, because it consists of more than 51 undecillion digits.

In Computer Science

This is the largest number that can accommodate a 128-bit signed integer signed int128 data type . An analogue of the 2038 problem for 128-bit computers will come no earlier than in an undecillion years (10 ³⁶ ) due to the large value of this number ^[2] .

In popular culture

In the film from the Futurama series - The Beast with a Billion Spins , this number is equal to the seventh double Mersenne number $M_{M_{7}}$ ${\ displaystyle M_ {M_ {7}}}$ , can be seen briefly in the "elementary proof of the Goldbach hypothesis ", and is known as the "martian prime".

Notes

↑ Sequence A000668 in OEIS (English)
↑ BBC: "The number glitch that can lead to catastrophe" // BBC, 5 May 2015 (eng.)

[1] Sequence A000668 in OEIS (English)

[2] BBC: "The number glitch that can lead to catastrophe" // BBC, 5 May 2015 (eng.)