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 618970019449562111 2 DO
11 107 162259276010288127 47 DC
12 127 170141183884105727 683 DC
13 521 686479766115057151 2 DO
14 607 531137992031728127 5 DO
15 1279 104079321168729087 20149 DC
16 2203 147597991697771007 19 DC
17 2281 446087557132836351 2 DO
18 3217 259117086909315071 2 DO
19 4253 190797007350484991 ? Pending
20 4423 285542542608580607 2 DO
21 9689 478220278225754111 2 DO
22 9941 346088282789463551 ? Pending
23 11213 281411201696392191 ? Pending
24 19937 431542479968041471 2 DO
25 21701 448679166511882751 2 DO
26 23209 402874115779264511 2 DO
27 44497 854509824011228671 2 DO
28 86243 536927995433438207 5 DO
29 110503 521928313465515007 5 DO
30 132049 512740276730061311 5 DO
31 216091 746093103815528447 ? Pending
32 756839 174135906544677887 ? Pending
33 859433 129498125500142591 ? Pending
34 1257787 412245773089366527 2 DO
35 1398269 814717564451315711 2 DO
36 2976221 623340076729201151 2 DO
37 3021377 127411683024694271 ? Pending
38 6972593 437075744924193791 2  
39 13466917 924947738256259071 ? Pending
40 20996011 125976895855682047 ? Pending
41 24036583 299410429733969407 2 DO
42 25964951 122164630577077247 ? Pending
43 30402457 315416475652943871 ? Pending
44 32582657 124575026053967871 ? Pending
45 37156667 202254406308220927 2 DO
46 42643801 169873516562314751 ? Pending
47 43112609 316470269697152511 ? Pending
48 57885161 581887266724285951 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

***

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.

***

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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