Problems & Puzzles: Puzzles

Puzzle 417. M(e) reversed primes. G. L. Honaker, Jr., asks for a Mersenne prime (greater than 31=2^5-1) such that its reversal remains prime. I made a fast search looking for Mersenne numbers (primes or composites) M(e)=2^e-1 such that M(e) reversed is prime (as a matter of fact only Strongpseudo prime) for e< 4369 and found that M(e) reversed is prime for e= 5, 53, 189, 293, 1107, 2181, 2695, & 2871. Q1.Can you find a Mersenne prime as asked for by Honaker? See http://primes.utm.edu/curios/page.php?curio_id=845 Q2. Please report any other e such that M(e) reversed is prime. Q3. What is the largest Mersenne prime reversal that has been successfully tested for primality?

---

J. K. Andersen wrote (May 08):

Q1 and Q3.
For the 41 known Mersenne primes 2^e-1 with e > 5, the reversal has been proven composite by a negative prp test or finding a factor. The 7 largest have a small factor. 7 divides the largest with e = 32582657. The largest performed prp test was at 909526 digits for e = 3021377 which gives no factor below 10^10. PRP tests were made by PrimeForm/GW.

Q2.
The reversal of 2^e-1 is a multidigit proven prime for e = 5, 53, 189, 293, 1107, 2181, 2695, 2871, and 7667. Primality proofs were made by Marcel Martin's Primo. It is a probable prime for e = 19999, 27471, 44537, 62323, 134367, 174295, and no others below 213000.

***

---