Palindromic Prime Exponents

Dobri · 2021-11-30, 06:07

Out of the total of 5,953 base-10 palindromic prime exponents < 10⁹, currently there are 1,889 remaining base-10 palindromic prime exponents for which the corresponding Mersenne numbers have no known factor (see the attached file). The list does not include the exponents for M2, M3, M5, and M7 which are the only four known Mersenne primes with base-10 palindromic prime exponents.

Dobri · 2021-11-30, 14:40

The initial post contains a list of base-10 palindromic prime exponents for which currently the corresponding Mersenne numbers have no known factor and are also of untested or unverified LL/PRP status.
In addition, this second post contains a shorter list of 292 exponents for which currently the corresponding Mersenne numbers have no known factor but are of verified C-LL/C-PRP status (see the attached file).

Dobri · 2022-03-06, 21:20

The following Wolfram language code generates the remaining palindromic prime exponents < 10⁹ for which the corresponding Mersenne numbers remain to be factored/verified.
Currently, there are 1880 such palindromic exponents within the range [100707001,..., 999676999].
Note that the code could be optimized for speed.

Code:

pp = PrimePi[10^9]; pn = 1;
count = 0; ic = 1; While[ic <= pp, pn = NextPrime[pn];
 If[(PalindromeQ[pn] == True) && (pn > 7), pns = ToString[pn];
  wppns = StringJoin["https://www.mersenne.org/report_exponent/?exp_lo=", pns, "&exp_hi=&text=1"];
  text = Import[wppns];
  fc = StringContainsQ[text, "Factored"];
  vc = StringContainsQ[text, "Verified"];
  If[(fc == False) && (vc == False), Print[pn]; count++;];
  ]; ic++;];
Print[count];

Dobri · 2022-09-04, 04:35

The smallest unfactored Mersenne numbers with palindromic exponents currently are
M15451, M16061, M17471, M31013, M35753, M37573, M38083, M74047, M77477, M78787, M96269, M97379, M98389, M1035301,...

Dobri · 2022-09-08, 16:57

M100707001 (Verified)
The smallest unverified Mersenne numbers with palindromic exponents currently are
M100767001, M101282101, M101343101, M101424101, M101474101, M101717101, M101777101, M101838101, M101919101, M101949101, M101999101, M102070201, M102232201, M102272201,...

The palindromic wavefront of untested Mersenne numbers with palindromic exponents is currently at
M129202921, M129484921, M129737921, M131252131, M133909331, M134535431, M134616431, M134919431, M135040531, M135161531, M135626531, M135646531, M135868531, M135929531,...

Dobri · 2022-10-01, 10:48

Below is a list of the bases in which the exponents of the known Mersenne primes are palindromic.

Code:

Exponent, Base
2, 3
3, 2
5, 2
7, 2
13, 3
17, 2
19, 18
31, 2
61, 6
89, 8
107, 2
127, 2
521, 11
607, 606
1279, 17
2203, 25
2281, 23
3217, 27
4253, 34
4423, 18
9689, 9688
9941, 30
11213, 59
19937, 41
21701, 41
23209, 50
44497, 67
86243, 60
110503, 13
132049, 107
216091, 223
756839, 127
859433, 144
1257787, 178
1398269, 231
2976221, 211
3021377, 344
6972593, 376
13466917, 2988
20996011, 406
24036583, 626
25964951, 1056
30402457, 341
32582657, 550
37156667, 2260
42643801, 353
43112609, 4223
57885161, 400
74207281, 1397
77232917, 610
82589933, 767

Only 7 exponents of Mersenne primes are palindromic in base 2:
3₁₀ = 11₂,
5₁₀ = 101₂,
7₁₀ = 111₂,
17₁₀ = 10001₂,
31₁₀ = 11111₂,
107₁₀ = 1101011₂, and
127₁₀ = 1111111₂.

Dr Sardonicus · 2022-10-01, 15:52

Quote:

Originally Posted by Dobri

Below is a list of the bases in which the exponents of the known Mersenne primes are palindromic.

Code:

Exponent, Base
2, 3
3, 2
5, 2
7, 2
13, 3
17, 2
19, 18
31, 2
61, 6
89, 8
107, 2
127, 2
521, 11
607, 606
<snip>

Any number is palindromic to a base larger than itself, since it only has one digit to that base. Any number n > 2 is palindromic to base n-1, n = 11_n-1.
Your list is also far from complete. Here is a listing for exponents up to 82589933.

Code:

Exponent base digits (expressed in decimal)
3 2 [1,1]
5 2 [1, 0, 1]
5 4 [1,1]
7 2 [1, 1, 1]
7 6 [1,1]
13 3 [1, 1, 1]
13 12 [1,1]
17 2 [1, 0, 0, 0, 1]
17 4 [1, 0, 1]
17 16 [1,1]
19 18 [1,1]
31 2 [1, 1, 1, 1, 1]
31 5 [1, 1, 1]
31 30 [1,1]
61 6 [1, 4, 1]
61 60 [1,1]
89 8 [1, 3, 1]
89 88 [1,1]
107 2 [1, 1, 0, 1, 0, 1, 1]
107 7 [2, 1, 2]
107 106 [1,1]
127 2 [1, 1, 1, 1, 1, 1, 1]
127 9 [1, 5, 1]
127 126 [1,1]
521 11 [4, 3, 4]
521 20 [1, 6, 1]
521 520 [1,1]
607 606 [1,1]
1279 17 [4, 7, 4]
1279 1278 [1,1]
2203 25 [3, 13, 3]
2203 31 [2, 9, 2]
2203 2202 [1,1]
2281 23 [4, 7, 4]
2281 38 [1, 22, 1]
2281 40 [1, 17, 1]
2281 2280 [1,1]
3217 27 [4, 11, 4]
3217 48 [1, 19, 1]
3217 3216 [1,1]
4253 34 [3, 23, 3]
4253 39 [2, 31, 2]
4253 4252 [1,1]
4423 18 [13, 11, 13]
4423 24 [7, 16, 7]
4423 34 [3, 28, 3]
4423 66 [1, 1, 1]
4423 4422 [1,1]
9689 9688 [1,1]
9941 30 [11, 1, 11]
9941 71 [1, 69, 1]
9941 9940 [1,1]
11213 59 [3, 13, 3]
11213 11212 [1,1]
19937 41 [11, 35, 11]
19937 47 [9, 1, 9]
19937 112 [1, 66, 1]
19937 19936 [1,1]
21701 41 [12, 37, 12]
21701 64 [5, 19, 5]
21701 124 [1, 51, 1]
21701 140 [1, 15, 1]
21701 21700 [1,1]
23209 50 [9, 14, 9]
23209 82 [3, 37, 3]
23209 23208 [1,1]
44497 67 [9, 61, 9]
44497 206 [1, 10, 1]
44497 44496 [1,1]
86243 60 [23, 57, 23]
86243 154 [3, 98, 3]
86243 160 [3, 59, 3]
86243 214 [1, 189, 1]
86243 86242 [1,1]
110503 13 [3, 11, 3, 11, 3]
110503 170 [3, 140, 3]
110503 110502 [1,1]
132049 107 [11, 57, 11]
132049 206 [3, 23, 3]
132049 262 [1, 242, 1]
132049 336 [1, 57, 1]
132049 132048 [1,1]
216091 223 [4, 77, 4]
216091 281 [2, 207, 2]
216091 343 [1, 287, 1]
216091 441 [1, 49, 1]
216091 216090 [1,1]
756839 127 [46, 117, 46]
756839 439 [3, 407, 3]
756839 756838 [1,1]
859433 144 [41, 64, 41]
859433 238 [15, 41, 15]
859433 721 [1, 471, 1]
859433 824 [1, 219, 1]
859433 859432 [1,1]
1257787 178 [39, 124, 39]
1257787 1257786 [1,1]
1398269 231 [26, 47, 26]
1398269 492 [5, 382, 5]
1398269 703 [2, 583, 2]
1398269 741 [2, 405, 2]
1398269 1398268 [1,1]
2976221 211 [66, 179, 66]
2976221 220 [61, 108, 61]
2976221 305 [31, 303, 31]
2976221 1041 [2, 777, 2]
2976221 2976220 [1,1]
3021377 344 [25, 183, 25]
3021377 738 [5, 404, 5]
3021377 1151 [2, 323, 2]
3021377 3021376 [1,1]
6972593 376 [49, 120, 49]
6972593 624 [17, 566, 17]
6972593 1903 [1, 1761, 1]
6972593 2519 [1, 249, 1]
6972593 6972592 [1,1]
13466917 2988 [1, 1519, 1]
13466917 13466916 [1,1]
20996011 406 [127, 152, 127]
20996011 602 [57, 563, 57]
20996011 3381 [1, 2829, 1]
20996011 3703 [1, 1967, 1]
20996011 3726 [1, 1909, 1]
20996011 3969 [1, 1321, 1]
20996011 4347 [1, 483, 1]
20996011 4410 [1, 351, 1]
20996011 20996010 [1,1]
24036583 626 [61, 211, 61]
24036583 1244 [15, 662, 15]
24036583 1816 [7, 524, 7]
24036583 24036582 [1,1]
25964951 1056 [23, 300, 23]
25964951 2259 [5, 199, 5]
25964951 3137 [2, 2003, 2]
25964951 4675 [1, 879, 1]
25964951 25964950 [1,1]
30402457 341 [261, 155, 261]
30402457 1950 [7, 1941, 7]
30402457 4088 [1, 3349, 1]
30402457 4891 [1, 1325, 1]
30402457 4958 [1, 1174, 1]
30402457 5402 [1, 226, 1]
30402457 30402456 [1,1]
32582657 550 [107, 391, 107]
32582657 3645 [2, 1649, 2]
32582657 3831 [2, 843, 2]
32582657 5416 [1, 600, 1]
32582657 32582656 [1,1]
37156667 2260 [7, 621, 7]
37156667 37156666 [1,1]
42643801 353 [342, 77, 342]
42643801 830 [61, 748, 61]
42643801 4770 [1, 4170, 1]
42643801 5300 [1, 2746, 1]
42643801 5364 [1, 2586, 1]
42643801 5400 [1, 2497, 1]
42643801 5724 [1, 1726, 1]
42643801 5960 [1, 1195, 1]
42643801 6360 [1, 345, 1]
42643801 42643800 [1,1]
43112609 4223 [2, 1763, 2]
43112609 43112608 [1,1]
57885161 400 [361, 312, 361]
57885161 518 [215, 377, 215]
57885161 986 [59, 533, 59]
57885161 1038 [53, 752, 53]
57885161 2679 [8, 175, 8]
57885161 5560 [1, 4851, 1]
57885161 7180 [1, 882, 1]
57885161 57885160 [1,1]
74207281 1397 [38, 33, 38]
74207281 1487 [33, 833, 33]
74207281 1534 [31, 821, 31]
74207281 1827 [22, 423, 22]
74207281 74207280 [1,1]
77232917 610 [207, 341, 207]
77232917 77232916 [1,1]
82589933 767 [140, 299, 140]
82589933 2727 [11, 289, 11]
82589933 4874 [3, 2323, 3]
82589933 82589932 [1,1]

Dobri · 2022-10-01, 17:22

Quote:

Originally Posted by Dr Sardonicus

Any number is palindromic to a base larger than itself, since it only has one digit to that base. Any number n > 2 is palindromic to base n-1, n = 11_n-1.
Your list is also far from complete. Here is a listing for exponents up to 21701.
...

Let me clarify that my post was concerned with the minimal bases b_min > 1 in which the exponents of Mersenne primes are palindromic.
Thanks for extending the list to all possible bases b up to b = n - 1.

Only 4 exponents of Mersenne primes are palindromic in a minimal base b_min = n - 1:

Code:

Exponent, Base
3, 2
19, 18
607, 606
9689, 9688

For completeness, let's note that all exponents are palindromic in base b = 1 (unary numeral system).

Dobri · 2022-12-29, 08:42

Palindromic Milestone Report: All 27-bit base-10 palindromic prime exponents less than 2²⁷ have been tested at least once.

These were the last remaining 27-bit ones: M129202921 (Verified), M129484921 (Unverified), M129737921 (Unverified), M131252131 (Verified), and M133909331 (Verified).

The next untested 28-bit base-10 palindromic prime exponent is M134919431.

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2022-09-04, 04:35	#4
Dobri "ม้าไฟ" May 2018 2²·5·23 Posts	The smallest unfactored Mersenne numbers with palindromic exponents currently are M15451, M16061, M17471, M31013, M35753, M37573, M38083, M74047, M77477, M78787, M96269, M97379, M98389, M1035301,...

2022-09-08, 16:57	#5
Dobri "ม้าไฟ" May 2018 2²·5·23 Posts	M100707001 (Verified) The smallest unverified Mersenne numbers with palindromic exponents currently are M100767001, M101282101, M101343101, M101424101, M101474101, M101717101, M101777101, M101838101, M101919101, M101949101, M101999101, M102070201, M102232201, M102272201,... The palindromic wavefront of untested Mersenne numbers with palindromic exponents is currently at M129202921, M129484921, M129737921, M131252131, M133909331, M134535431, M134616431, M134919431, M135040531, M135161531, M135626531, M135646531, M135868531, M135929531,...

2022-12-29, 08:42	#9
Dobri "ม้าไฟ" May 2018 2²×5×23 Posts	Palindromic Milestone Report: All 27-bit base-10 palindromic prime exponents less than 2²⁷ have been tested at least once. These were the last remaining 27-bit ones: M129202921 (Verified), M129484921 (Unverified), M129737921 (Unverified), M131252131 (Verified), and M133909331 (Verified). The next untested 28-bit base-10 palindromic prime exponent is M134919431.