The even perfect numbers are closely related to the Mersenne primes. We currently know $51$ Mersenne primes and hence $51$ perfect numbers.
It has already been checked for which of those perfect numbers $P$ , the number $P+1$ is prime. The formula for $P$ is
$$P=2^{n-1}(2^n-1)$$ where $n$ is an exponent for which $2^n-1$ is prime.
The exponents $n=2,3,13,19$ are known to give a prime. For all other exponents non-trivial factors are known except of the $49$ th exponent $n=74207281$
Has this case been checked by someone ? If not , I invite everyone to search a nontrivial prime factor.
According to my calculations , there is no prime factor below $2\cdot 10^{10}$ , nevertheless the chance of this number to be prime is extremely small.
As reported by mersenneforum user Neptune just yesterday, this number is divisible by the $17$-digit prime $14344999215792989$, found using the elliptic curve method. (Link posted by Martin Hopf in the comments.)