Prime number of Mersenne

In Mathematical and more precisely in Arithmetic modular, a prime number of Mersenne is a Prime number being written in the form 2^p - 1, p being first. These prime numbers owe their name with a scholar and mathematician French of the 17th century, Marin Mersenne.

More generally, the numbers of Mersenne (not necessarily first, but candidates with the being) are the numbers of the form 2^p - 1 , with p first. One uses the notation M_p = 2^p - 1 .

More the small numbers first of Mersenne are:

• 2²-1

• 2³-1

• 31 = 2⁵-1
• 127 = 2⁷-1
• But 2047 = 2¹¹-1 = 23 X 89 is a number of Mersenne, but not first.

It is shown that an entirety of the form 2ⁿ-1 cannot be first if N is not itself first. Thus 2⁴-1=15 is not Mersenne, nor first.

Properties of the numbers of Mersenne

The numbers of Mersenne have the following properties:

• If $N$ is first (for example the product $N\; =\; R\; S$ where neither $r$, nor $s$ is not equal to 1) then the number of Mersenne $2^\; \{R\; S\}\; -\; 1$ is not first.

Indeed, by noticing that the continuation of the $r$ first terms of the geometrical Suite $(2^s)\; \_\; \{S\; \backslash \; Ge\; 0\}$ is equal to: $\backslash \; displaystyle\; 1+2^s+\; (2^\; \{S\})\; ^2+\dots \; +\; (2^\; \{S\})\; ^\; \{r-1\}\; =\; \backslash \; frac\; \{2^\; \{R\; S\}\; -\; 1\}\; \{2^s\; -1\}$, one proves that $\backslash \; displaystyle\; 2^\; \{R\; S\}\; -\; 1$ is divisible by $\backslash \; displaystyle\; 2^s\; -1$ which is different from 1 as soon as S is also distinct from 1.

Note: one can also use the formula $\backslash \; displaystyle\; 1+2^r+\; (2^\; \{R\})\; ^2+\dots \; +\; (2^\; \{R\})\; ^\; \{s-1\}\; =\; \backslash \; frac\; \{2^\; \{R\; S\}\; -\; 1\}\; \{2^r\; -1\}$, to prove this result.

Thus, when one seeks prime numbers via the numbers of Mersenne, one knows already that it is necessary to avoid the candidates like $\backslash \; displaystyle\; 2^4\; -1$ (i.e. 15), $\backslash \; displaystyle\; 2^6\; -1$ (i.e. 63) or $\backslash \; displaystyle\; 2^9\; -1$ (i.e. 511 = $7\; \backslash \; times\; 73$).

The idea is now to sharpen the selection criteria of the prime numbers $\backslash \; displaystyle\; p$…

• M_n is the sum of binomial coefficients minus 1: $M\_n\; =\; \backslash \; sum\_\; \{i=0\}\; ^\; \{N\}\; \{N\; \backslash \; choose\; I\}\; -\; 1$.

• If has divides M_q (Q first) then has has the following properties: has = 1 (MOD 2q) and: has = + 1 (MOD 8).

• a theorem of Euler involves that: M_q (Q first) is first if and only there exists a single pair $(X,\; there)$ such as: $M\_q\; =\; \{(2x)\}^2\; +\; 3\; \{(3y)\}^2$ with Q >= 5 . Very recently, Bas Jansen studied $M\_q\; =\; x^2\; +\; dy^2$ for d=0..48 .

• Is Q = 3 (MOD 4) first. $2q+1$ is also first if and only if: 2q+1 divides M_q .

• Reix recently showed that the numbers of Mersenne M_q (Q first > 3), first or not, are written: $M\_q\; =\; \{(8x)\}^2\; -\; \{(3qy)\}^2\; =\; \{(1+Sq)\}^2\; -\; \{(Dq)\}^2$. Obviously, if the pair (X, there) is single, then M_q is first.

• Ramanujan showed that the equation: $M\_q\; =\; 6+x^2$ has only 3 solutions with Q first: 3,5, and 7 (and 2 solutions with Q not-first).

• All the factors first of a number of Mersenne associated with the prime number p are form kp+1 where K is a natural entirety. Two distinct numbers of Mersenne are always first between-them.

History

The prime numbers of Mersenne are related to the perfect numbers, which are the numbers equal to the sum of their own dividers. It is this connection which historically justified the study of the prime numbers of Mersenne. As of the fourth century BC, Euclide showed that if M = 2^p - 1 is a prime number, then M (M+1) /2 = 2 ^(p-1) (2^p - 1) is a Perfect number. Two millenia later, at the 17th century, Euler proved that all the perfect numbers even have this form. No odd perfect number is known, and it is supposed that there is none.

M_a divides M_p if has divides p . Thus so that M_p is first, it is necessary that p is first. That simplifies already the search for prime numbers of Mersenne. The reciprocal one is not true: M_p can be made up whereas p is first; the smallest example is 2¹¹-1 = 23×89.

For the numbers of Mersenne there exists a method (comparatively) very fast to determine if they are first, developed originally by Lucas in 1878 and improved by Lehmer in the Années 1930. One can indeed show that for p prime number odd $M\_p\; =\; 2^p-1$ is first if and only if $M\_p$ divides $S\_\; \{p-1\}$, where $S\_1\; =\; 4$ and for K > 1, $S\_\; \{k+1\}\; =\; S\_\; \{K\}\; ^2\; -2$.

Mersenne did not invent the numbers of Mersenne, but it provided a list of prime numbers of Mersenne to exhibitor 257. Unfortunately this list was false: it included by error 67 and 257, and omitted 61,89 and 107.

The first four prime numbers of Mersenne were known as of Antiquity. The fifth (2¹³-1) was discovered before 1461 by an unknown. The two following was found by Cataldi in 1588. More than one century later, in 1750, Euler still found one of them. The following in the chronological order (but nonnumerical) was found by Lucas in 1876, then one by Pervushin in 1883. Two others were found at the beginning of the 20th century by Powers in 1911 and 1914.

Research for the prime numbers of Mersenne was revolutionized by the introduction of the electronic computer electronic computers. The first identification of a number of Mersenne by this means took place at 10 p.m. the January 30th 1952 by a computer SWAC at the Institute of Numerical Analysis ( Institute for Numerical Analysis ) of the campus of the University of California - Los Angeles, under the direction of D.H. Lehmer, with a program written by R.M. Robinson.

It had been the first prime number of Mersenne identified for 38 years. The following was found less than two hours later by the same computer, which found of them three of more in the next months.

In September 2006, 44 prime numbers of Mersenne were known, and the greatest prime number known was a prime number of Mersenne, 2^{: 32582657}-1. Like several of its predecessors, he was discovered by a Calcul distributed under the aegis of the project GIMPS, Great Internet Mersenne Prime Search (which means “great research by Internet of prime numbers of Mersenne”).

List

In September 2006, 44 prime numbers of Mersenne M_p=2^p-1 were known.

^* One does not know if there exist or not one or more prime numbers of Mersenne not yet discovered between the 39e (M_{: 13466917}) and the 44e (M_{3: 2582657}). This classification is thus provisional.

See too

Internal bonds

• GIMPS
• List of the great numbers
• Prime number of Fermat

External bonds

• http://www.utm.edu/research/primes/mersenne.shtml
• Banner page of project GIMPS
• Discovered of the 42 {{E}}
• Slashdot - Discovered of the 42 {{E}}
• M_q = (8x) ² - (3qy) ² Proof
• Numbers of Mersenne
• Numbers of Mersenne first
• M_q = x² + d.y² Thesis

Simple: Mersenne precedes

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