Find primes p and q such that p/(p+1) + (q+1)/q = 2n/(n+2) where n is a natural number
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Answer:
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.
Step-by-step explanation:
The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).
If n is a composite number then so is 2n − 1. (2ab − 1 is divisible by both 2a − 1 and 2b − 1.) This definition is therefore equivalent to the definition as a prime number of the form Mp = 2p − 1 for some prime p.
More generally, numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89.
Mersenne primes Mp are also noteworthy due to their connection to perfect numbers.
As of April 2020, 51 Mersenne primes are known. The largest known prime number, 282,589,933 − 1, is a Mersenne prime.[1] Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project on the Internet.
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