梅森數是指形如2n−1的數, 其記號為Mn;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 Mn; if a Mersenne number is prime it is called Mersenne prime. If n of Mn is not a prime, the Mersenne number definitely not a Mersenne prime. For example: If x = 123456789123456789123456789 then 2x-1 is not a prime. Proof: x is not a prime, and it is dividable by 3. Therefore, 2x-1 = (2^y)3 – 1, (that 2^y = 241152263041152263041152263 ) -> (2^y)3 – 1 = (2^y – 1) [(2^y) 2 – 2(2^y) +1], therefore, 2123456789123456789123456789-1 can be dividable by (241152263041152263041152263– 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.