啟蒙數學

梅森數 M_p=2^p-1是素數的條件是(1) p 是素數 (2) M_p不能被任何2kp + 1這種型的數整除,其中k是整數且2kp + 1 < 2^p-1. 它的孿生兄弟K_x=2^x+1,其中x=2ⁿ. 它是素數的條件是K_x不能被任何2ix + 1這種型的數整除,其中i是整數且2ix + 1 < 2^x+1. 例如K₁=2²+1=5, K₂=2⁴+1=17, K₃=2⁸+1=257, K₄=2¹⁶+1=65537都是素數. 但K₅與K₆不是素數. K₅=2³²+1= 4294967297 = 641 * 6700417. 641 = (2)(32)(10)+1; x = 32 與 i = 10. 6700417 = (2)(32)(104694)+1; x = 32與 i = 104694. K₆=2⁶⁴+1= 18446744073709551617 = 274177 * 67280421310721. 274177 = (2)(64)(2142)+1; x = 64 與 i = 2142. 67280421310721 = (2)(64)(525628291490)+1; x = 64與 i = 525628291490. 它比梅森數的優點是不需要確定P是素數.

你知道下面的梅森數, M_1567271, M_1567259, M_1567103, M_1567031, and M_1566923,有下列的3134543, 3134519, 3134207, 3134063和3133847對應因子嗎？上例中的數字都是素數. 使得找的該數們的因子有其困難的程度. 尤其是當素數的值大到一定程度以上如前面所提的例子. 找他們的因子就很難了,甚至於用電腦來找. 相較下，2^ab – 1, 其ab不是素數, 證明它不是素數且有 (2^a– 1) 和(2^b– 1) 的因子就比較容易了. Do you know the following Mersenne Numbers, M_1567271, M_1567259, M_1567103, M_1567031, and M_1566923, have factors as 3134543, 3134519, 3134207, 3134063 and 3133847, respectively?  The above numbers are all primes that contribute the difficulty nature for people to find their cofactors.  When the values of the prime numbers become large to some extend as the aforementioned examples, it is hard to obtain their cofactors, even with the aid of a computer. On the other hand, it is relatively easier to prove and find the cofactor of M_composite that has the form of 2^ab – 1. Note that ab is not a prime.  The following is one of the proofs of 2^ab – 1 that shows 2^ab – 1 has (2^a– 1) and (2^b– 1) as its factors.

梅森數是指形如2ⁿ−1的數, 其記號為M_n;如果一個梅森數是質(素)數它被稱為梅森質(素)數.  如n 不是質(素)數則其梅森數絕對不是梅森質(素)數. 例子請參閱下列藍色部份的證明. 如n是質(素)數則梅森數有可能是梅森質(素)數.  以下是n是質(素)數其梅森數卻不是梅森質(素)數的例子. 例如2¹¹−1 被23整除; 2²³−1 被47整除;2³³¹⁰³¹−1 被662063 整除. 如需更多例子請參閱下列表格.A Mersenne number is the number of the form 2ⁿ-1, the symbol for it is M_n; if a Mersenne number is prime it is called Mersenne prime. If n of M_n is not a prime, the Mersenne number definitely not a Mersenne prime. For example: If x = 123456789123456789123456789  then 2^x-1 is not a prime.  Proof: x is not a prime, and it is dividable by 3.  Therefore,  2^x-1 = (2^y)³ – 1, (that 2^y = 2^{41152263041152263041152263} )  -> (2^y)³ – 1 = (2^y – 1) [(2^y) ² – 2(2^y) +1], therefore, 2^{123456789123456789123456789}-1 can be dividable by (2^{41152263041152263041152263}– 1). If n is a prime, the Mersenne number could be a prime. The following is n is prime for a Mersenne number but  it is not a Mersenne prime.  For example 2¹¹−1 is dividable by 23; 2²³−1 is dividable by 47; 2³³¹⁰³¹−1 is dividable by 662,063. For more examples please refer to the following table.