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Questions in category: 初等数论 (Elementary Number Theory).

目前已知的梅森素数(Mersenne primes)

Posted by haifeng on 2019-09-21 09:47:10 last update 2019-09-21 15:05:06 | Answers (0) | 收藏


形如 $2^n-1$ 的数, 如果是素数, 则称为梅森素数(Mersenne prime).

由于当 $n$ 是合数时, 比如 $n=pq$, 则 $2^{pq}-1$ 总可以分解. 具体的, 若 $q=2k$, 则有 $2^{2pk}-1=(2^{pk}-1)(2^{pk}+1)$. 若 $q=2k+1$, 则含有因子 $2^p-1$. 因此 $2^n-1$ 称为素数的必要条件是 $n$ 是素数.

一般记梅森素数为 $M_p=2^p-1$, 其中 $p$ 为素数. 目前已知的梅森素数有

No. $p$ $M_p$ isPrime
1 2 $2^2-1=3$ Y
2 3 $2^3-1=7$ Y
3 5 $2^5-1=31$ Y
4 7 $2^7-1=127$ Y
? 11 $2^{11}-1=2047$ N
5 13 $2^{13}-1=8191$ Y
6 17 $2^{17}-1=131071$ Y
7 19 $2^{19}-1=524287$ Y
? 23 $2^{23}-1=8388607$ N
? 29 $2^{29}-1=536870911$ N
8 31 $2^{31}-1=2147483647$ Y

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以下仅列出梅森素数

No. $p$ $M_p$ Value
9 61 $2^{61}-1$ 2305843009213693951
10 89 $2^{89}-1$ 618970019642690137449562111
11 107 $2^{107}-1$ 162259276829213363391578010288127
12 127 $2^{127}-1$ 170141183460469231731687303715884105727
13 521 $2^{521}-1$ ?
14 607 $2^{607}-1$ ?
15 1279 $2^{1279}-1$ ?
16 2203 $2^{2203}-1$ ?
17 2281 $2^{2281}-1$ ?
18 3217 $2^{3217}-1$ ?
19 4253 $2^{4253}-1$ ?
20 4423 $2^{4423}-1$ ?

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使用 Calculator.exe 验证: isprime(2^3217-1)

>> isprime(2^3217-1)
in> isprime(2^3217-1)
in> 2^3217-1

开元棋牌优惠out> 259117086013202627776246767922441530941818887553125427303974923161874019266586362086201209516800483406550695241733194177441689509238807017410377709597512042313066624082916353517952311186154862265604547691127595848775610568757931191017711408826252153849035830401185072116424747461823031471398340229288074545677907941037288235820705892351068433882986888616658650280927692080339605869308790500409503709875902119018371991620994002568935113136548829739112656797303241986517250116412703509705427773477972349821676443446668383119322540099648994051790241624056519054483690809616061625743042361721863339415852426431208737266591962061753535748892894599629195183082621860853400937932839420261866586142503251450773096274235376822938649407127700846077124211823080804139298087057504713825264571448379371125032081826126566649084251699453951887789613650248405739378594599444335231188280123660406262468609212150349937584782292237144339628858485938215738821232393687046160677362909315071
2^3217-1 is a Mersenne number, we use Lucas-Lehmer test.
2^3217-1 is a Mersenne prime.

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用时大约3分钟

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References:

https://www.mersenne.org/primes/

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