Problems & Puzzles: Puzzles

Puzzle 860. Primes of the Mersene primes reversed.

In the Curio's page by G. L. Honaker, Jr. and Chris Caldwell, you may read that "31 is the only known Mersenne emirp"

As a matter if fact, nowadays, the only Mersene primes that remain primes when reversed are the first three of them: 3, 7 & 13. Are there more of these in the rest of the 46 known up today?

For the purpose of this puzzle we will call: Mn=Mersenne prime. Mn=2^n-1;
R(Mn)=Reverse of Mn and SPFR(Mn)=the smallest prime factor of R(Mn)

By direct observation of the following table we may discard 25 other Mersenne prime numbers as candidate for primes when reversed.

By direct calculation in a code that I made in Ubasic, I have discarded other 6 Mersenne prime numbers as candidate for primes when reversed.

Accordingly there are only 46-25-6=15 Mersenne prime numbers as candidate for primes when reversed, other than the first three.

These 15 n values for M(n)=2^n-1, are: 4253, 9941, 11213, 216091, 756839, 859433, 3021377, 13466917, 20996011, 25964951, 30402457, 32582657, 42643801, 43112609 & 74207281.

All this is shown in the following table:

 Mn = 2^n-1 R(Mn)= reverse of Mn SPFR(Mn) = smallest prime factor of R(Mn) DC = Direct Calculation by CR D0= Direct observation of Table # n Mn SPFR(Mn) Comment 1 2 3 3 Prime 2 3 7 7 Prime 3 5 31 13 Prime 4 7 127 7 DC 5 13 8191 2 DO 6 17 131071 13 DC 7 19 524287 5 DO 8 31 2147483647 2 DO 9 61 2.30584E+18 2 DO 10 89 618970019…449562111 2 DO 11 107 162259276…010288127 47 DC 12 127 170141183…884105727 683 DC 13 521 686479766…115057151 2 DO 14 607 531137992…031728127 5 DO 15 1279 104079321…168729087 20149 DC 16 2203 147597991…697771007 19 DC 17 2281 446087557…132836351 2 DO 18 3217 259117086…909315071 2 DO 19 4253 190797007…350484991 ? Pending 20 4423 285542542…608580607 2 DO 21 9689 478220278…225754111 2 DO 22 9941 346088282…789463551 ? Pending 23 11213 281411201…696392191 ? Pending 24 19937 431542479…968041471 2 DO 25 21701 448679166…511882751 2 DO 26 23209 402874115…779264511 2 DO 27 44497 854509824…011228671 2 DO 28 86243 536927995…433438207 5 DO 29 110503 521928313…465515007 5 DO 30 132049 512740276…730061311 5 DO 31 216091 746093103…815528447 ? Pending 32 756839 174135906…544677887 ? Pending 33 859433 129498125…500142591 ? Pending 34 1257787 412245773…089366527 2 DO 35 1398269 814717564…451315711 2 DO 36 2976221 623340076…729201151 2 DO 37 3021377 127411683…024694271 ? Pending 38 6972593 437075744…924193791 2 39 13466917 924947738…256259071 ? Pending 40 20996011 125976895…855682047 ? Pending 41 24036583 299410429…733969407 2 DO 42 25964951 122164630…577077247 ? Pending 43 30402457 315416475…652943871 ? Pending 44 32582657 124575026…053967871 ? Pending 45 37156667 202254406…308220927 2 DO 46 42643801 169873516…562314751 ? Pending 47 43112609 316470269…697152511 ? Pending 48 57885161 581887266…724285951 5 DO 49 74207281 300376418084...391086436351 ? Pending

Q1. Find if R(Mn) are prime numbers (probably prime) or find its Small Prime Factor, for these 15 n values given above with a "pending" comment.

Contribution came from Emmanuel Vantieghem and Jan van Delden

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Emmanuel wrote:

Here is what I could find about puzzle 860 :
I list {p, SPFR(Mn) } :
{ 4253, 2399 }
{ 9941, 15383 }
{ 11213, 1613 }
{ 216091, > 3*109 }. R(Mn) is defintely composite (Fermat base 2 test)
{ 756839, 21683 }
{ 859433, 1311241 }
{ 3021377, > 10^9 } Fermat base 2 test is still running
{ 13466917, 7 }
{ 20996011, 113 }
{ 25964951, 47387 }
{ 30402457, 13 }
{ 32582657, 7 }
{ 42643801, 20323427}
{ 43112609, 47 }
{ 74207281, 1499 }
The test for 3021377 probably will take about two weeks (if no error messages will pop up).
But I take the risk to test this.

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Jan van Delden wrote:

RM(4253)     divisor 2399

RM(9941)     divisor 15383

RM(11213)    divisor 1613

RM(216091)   divisor 5367143153

RM(756839)   divisor 21683

RM(859433)   divisor 1311241

RM(13466917) divisor 7

RM(20996011) divisor 113

RM(25964951) divisor 47387

RM(30402457) divisor 13

RM(32582657) divisor 7

RM(42643801) divisor 20323427

RM(43112609) divisor 47

RM(74207281) divisor 1499

RM(3021377) status unknown

A Fermat test would take about 8 days.

I decided to try factoring. No divisor < 1.4*10^10, trial division. No result upon using Pollard Rho (after 27 hours).

***

Emmanuel wrote on Dec 31, 2016:

Just a bit after midnight my PC announced that  Rev(2^3021377 - 1)  is not prime !
The Fermat base 2 test took a bit more than 13 days with Mathematica.

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