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This page contains Mersenne-related numbers.

List of Mersenne-related numbers Edit

127 (one hundred twenty-seven) is a positive integer equal to \(2^{2^3-1}-1\). It is notable in computer science for being the maximum value of an 8-bit signed integer. It is the 4th Mersenne prime.

The Lucas–Lehmer primality test, which is used for finding the largest known primes, gives 194 after two iterations.

496 (four hundred ninety-six) is the third perfect number.[1] Its divisors are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496.

2,047 is the smallest composite Mersenne number with prime index, in this case, (211−1). The next Mersenne number however, which is 213−1 or 8,191, is prime.

It is also the smallest strong pseudoprime to base 2.

In the fast-growing hierarchy, it is equal to f2(8)−1 and f3(2)−1.

The number 13 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Wagstaff prime is equal to (213+1)/3 = 2,731.

8,128 (eight thousand one hundred twenty-eight) is the fourth perfect number.[2]

\(8,191=2^{13}-1\) is the smallest Mersenne prime which is not an exponent of another Mersenne prime.

It is also the largest known number which is a repunit with at least three digits in more than one base. The Goormaghtigh conjecture states that 31 and 8,191 are the only two numbers with this property.

The number 13 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture.

The number 17 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Wagstaff prime is equal to (217+1)/3.

Its decimal expansion is 43,691.

The number 17 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Mersenne prime is equal to 217−1 or M17. It is also the 6th known Mersenne prime.

Its decimal expansion is 131,071.

The number 19 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Wagstaff prime is equal to (219+1)/3.

Its decimal expansion is 174,763.

The number 19 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Mersenne prime is equal to 219−1 or M19. It is also the 7th known Mersenne prime.

Its decimal expansion is 524,287.

33,550,336 (thirty-three millions five hundred fifty thousands three hundred thirty-six) is the fifth perfect number.[3]

62,914,441 is the smallest prime factor of the composite double Mersenne number MM19 or M524,287.

231,733,529 is the smallest prime factor of the composite double Mersenne number MM17.

The number 31 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Wagstaff prime is equal to (231+1)/3.

Its decimal expansion is 715,827,883.

2,147,483,647 is a positive integer equal to \(2^{31} - 1 = 2^{2^5 - 1} - 1\). It is notable in computer science for being the maximum value of a 32-bit signed integer, which have the range [-2147483648, 2147483647]. It is also a prime number (conveniently for cryptographers), and so the 8th Mersenne prime.

Its full name in English is "two billion/milliard one hundred forty-seven million four hundred eighty-three thousand six hundred forty-seven," where the short scale uses "billion" and the long scale uses "milliard."

The number 31 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture.

8,589,869,056 is the sixth perfect number.[4] Furthermore, it is the largest known perfect number not containing digit '4'.

338,193,759,479 is the smallest prime factor of the composite double Mersenne number MM13.

295,257,526,626,031 is the smallest prime factor of the composite double Mersenne number MM31.

9,007,199,254,740,991 is a positive integer equal to \(2^{53} - 1\). It is notable in computer science for being the largest odd number which can be represented exactly in the double floating-point format (which has a 53-bit significand).

Its prime factorization is 9,007,199,254,740,991 = 6,361 × 69,431 × 20,394,401.

The number 61 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture. The corresponding Wagstaff prime is equal to (261+1)/3.

Its decimal expansion is 768,614,336,404,564,651.

\(2^{107}-1\) is the largest known Mersenne prime not containing the digit '4'. Its full decimal expansion is 162,259,276,829,213,363,391,578,010,288,127.

It has been conjectured, that no number larger than 127 holds all three conditions of the New Mersenne conjecture. The corresponding Wagstaff prime is equal to (2127+1)/3.

Its decimal expansion is 56,713,727,820,156,410,577,229,101,238,628,035,243.

It has been conjectured, that no number larger than 127 holds all three conditions of the New Mersenne conjecture. The corresponding Mersenne prime is equal to 2127−1.

Its decimal expansion is 170,141,183,460,469,231,731,687,303,715,884,105,727.

\(2^{521}-1=512*2^{512}-1 \approx 6.8647976601306097 \times 10^{156}\) is the largest known Mersenne prime which is also a Woodall number. There are no other such numbers smaller than \(2^{549,755,813,927}-1\).

Its full decimal expansion is 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151.

In the fast-growing hierarchy, it is equal to f2(512)−1.

Sources Edit

  1. Wikipedia article
  2. Wikipedia article
  3. Wikipedia article
  4. Wikipedia article