galois.mersenne_exponents¶

galois.mersenne_exponents(n=None)[source]¶

Returns all known Mersenne exponents \(e\) for \(e \le n\).

A Mersenne exponent \(e\) is an exponent of \(2\) such that \(2^e - 1\) is prime.

Parameters: n (int, optional) – The max exponent of 2. The default is None which returns all known Mersenne exponents.
Returns: The list of Mersenne exponents \(e\) for \(e \le n\).
Return type: list

References

https://oeis.org/A000043

Examples

# List all Mersenne exponents for Mersenne primes up to 4000 bits
In [1]: e = galois.mersenne_exponents(4000); e
Out[1]: 
[2,
 3,
 5,
 7,
 13,
 17,
 19,
 31,
 61,
 89,
 107,
 127,
 521,
 607,
 1279,
 2203,
 2281,
 3217]

# Select one Merseene exponent and compute its Mersenne prime
In [2]: p = 2**e[-1] - 1; p
Out[2]: 259117086013202627776246767922441530941818887553125427303974923161874019266586362086201209516800483406550695241733194177441689509238807017410377709597512042313066624082916353517952311186154862265604547691127595848775610568757931191017711408826252153849035830401185072116424747461823031471398340229288074545677907941037288235820705892351068433882986888616658650280927692080339605869308790500409503709875902119018371991620994002568935113136548829739112656797303241986517250116412703509705427773477972349821676443446668383119322540099648994051790241624056519054483690809616061625743042361721863339415852426431208737266591962061753535748892894599629195183082621860853400937932839420261866586142503251450773096274235376822938649407127700846077124211823080804139298087057504713825264571448379371125032081826126566649084251699453951887789613650248405739378594599444335231188280123660406262468609212150349937584782292237144339628858485938215738821232393687046160677362909315071

In [3]: galois.is_prime(p)
Out[3]: True

# Display all known Mersenne exponenets
In [4]: galois.mersenne_exponents()
Out[4]: 
[2,
 3,
 5,
 7,
 13,
 17,
 19,
 31,
 61,
 89,
 107,
 127,
 521,
 607,
 1279,
 2203,
 2281,
 3217,
 4253,
 4423,
 9689,
 9941,
 11213,
 19937,
 21701,
 23209,
 44497,
 86243,
 110503,
 132049,
 216091,
 756839,
 859433,
 1257787,
 1398269,
 2976221,
 3021377,
 6972593,
 13466917,
 20996011,
 24036583,
 25964951,
 30402457,
 32582657,
 37156667,
 42643801,
 43112609]