Problems & Puzzles: Puzzles

Puzzle 858. Updating the Puzzle 203. 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. As  matter 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. These integers, 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 Now (December 3, 2016), eight years later, we ask to complete the work until the 49th Mersenne primes nowadays already known (Dec 2016). Q1. Are the RPP corresponding to the M(p) 45, 46, ..., 49 primes or  not.? If not, give us its smallest prime factor?

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Contribution came from Emmanuel Vantieghem

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Emmanuel wrote, on Dec 9, 2016:

This is what I found about puzzle 858 :
 

RPP(42643801)  is divisible by  3593
 

RPP(37156667), RPP(43112609)  and  RPP(57885161)  are divisible by 7.
 

RPP(74207281)  has no divisor < 3*10^10.

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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 the computations)

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 I used Mathematica for all computations..

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