Problems & Puzzles:
Puzzle 858. Updating the
Here we ask if
the integers one unit more than the perfect number corresponding to
the Mersenne prime numbers 45th,...49th are primes or not. If they
are not primes, then we ask for its smallest prime factor.
of fact this was the subject of our now old
Puzzle 203, so smartly solved negatively for several puzzler
for the state of known primes until May 2008. These days the largest
known Mersenne prime number
was the 44th.
one unit more than a perfect number, were called in that
puzzle, "right perfect primes" (RPP), and the only four know, now
and then, were 7, 29, 33550337 & 137438691329,
discovered by Labos E. See A061644
3, 2016), eight
years later, we ask to complete the work until the 49th Mersenne
primes nowadays already known (Dec 2016).
Are the RPP corresponding to the M(p) 45, 46, ..., 49 primes
or not.? If not, give us its smallest prime factor?
Contribution came from Emmanuel Vantieghem
Emmanuel wrote, on Dec 9, 2016:
This is what I found about puzzle
RPP(42643801) is divisible by
RPP(37156667), RPP(43112609) and
RPP(57885161) are divisible by 7.
RPP(74207281) has no divisor <
Finding 3539 was simple : trial division !
No divisor result : idem.
The search was facilitated by the fact
that every prime divisor p of an RPP that is not divisible
by 7 must be of the form 7k+1, 2 or 4. (Indeed, an RPP is
of the form 2x2 - x +1 : this quadratic polynomial should
have a root modulo p and thus, it'x discriminant (= -7 )
must be a quadratic residue modulo p).
I have not tested m =
RPP(742072810) to be composite. I just said that it has no
divisor < 3*10^10. It can be prime. I tried to compute
2^(m-1) modulo m but, with my PC, it would take about 50
years to get a result. (I would be dead before the end of
I used Mathematica for