Left and Right Truncatable Triprimes

What was done with left and right truncatable semiprimes can also be done with triprimes. The very first triprime is 8 = 2 x 2 x 2 and, by adding suitable digits to the LEFT, we get (permalink):

Triprime             | Factorisation
---------------------------------------------
8                    | 2 * 2 * 2
18                   | 2 * 3 * 3
318                  | 2 * 3 * 53
2318                 | 2 * 19 * 61
12318                | 2 * 3 * 2053
112318               | 2 * 89 * 631
6112318              | 2 * 101 * 30259
16112318             | 2 * 53 * 152003
216112318            | 2 * 7687 * 14057
4216112318           | 2 * 11 * 191641469
24216112318          | 2 * 73 * 165863783
124216112318         | 2 * 19 * 3268845061
5124216112318        | 2 * 3203 * 799908853
25124216112318       | 2 * 3 * 4187369352053
225124216112318      | 2 * 431 * 261164983889
2225124216112318     | 2 * 193 * 5764570508063

Next we have 12 = 2 x 2 x 3 which gives (permalink):

Triprime             | Factorisation
---------------------------------------------
12                   | 2 * 2 * 3
212                  | 2 * 2 * 53
5212                 | 2 * 2 * 1303
15212                | 2 * 2 * 3803
315212               | 2 * 2 * 78803
2315212              | 2 * 2 * 578803

The next triprime (18) gives the same chain as did 8. 

Let's try adding digits to the RIGHT this time. Starting with 8 we don't get very far as no digit when adding to the right of 8 will produce a triprime. However, with 12 we have more success (permalink):

Triprime             | Factorisation
---------------------------------------------
12                   | 2 * 2 * 3
124                  | 2 * 2 * 31
1244                 | 2 * 2 * 311
12445                | 5 * 19 * 131
124453               | 7 * 23 * 773
1244534              | 2 * 349 * 1783
12445341             | 3 * 167 * 24841
124453414            | 2 * 23 * 2705509
1244534143           | 19 * 19 * 3447463
12445341431          | 769 * 2053 * 7883
124453414317         | 3 * 23789 * 1743851
1244534143173        | 3 * 31 * 13382087561
12445341431731       | 13 * 887 * 1079294201

The next triprime (18) produces this chain (permalink):

Triprime             | Factorisation
---------------------------------------------
18                   | 2 * 3 * 3
182                  | 2 * 7 * 13
1825                 | 5 * 5 * 73
18255                | 3 * 5 * 1217
182553               | 3 * 7 * 8693
1825534              | 2 * 227 * 4021
18255345             | 3 * 5 * 1217023
182553451            | 31 * 331 * 17791
1825534511           | 17 * 967 * 111049
18255345115          | 5 * 7 * 521581289
182553451151         | 17 * 5147 * 2086349
1825534511511        | 3 * 53 * 11481349129

With a large triprime like 28282 = 2 x 79 x 179 we end up with (permalink):

Triprime             | Factorisation
---------------------------------------------
28282                | 2 * 79 * 179
282821               | 7 * 11 * 3673
2828211              | 3 * 619 * 1523
28282113             | 3 * 3 * 3142457
282821134            | 2 * 5099 * 27733
2828211341           | 17 * 29 * 5736737
28282113411          | 3 * 32621 * 288997
282821134111         | 43 * 563 * 11682479
2828211341111        | 13 * 26591 * 8181517
28282113411116       | 2 * 2 * 7070528352779
282821134111163      | 59 * 1307821 * 3665317

So the program is working well.

Left Truncatable Semiprimes

Just as with primes, we can have left truncatable and right truncatable semiprimes. The smallest semiprime is 6 = 2 x 3 and using that as our starting point, we can build a chain of left truncatable semiprimes as shown below (permalink):

Semiprime                | Factorisation
---------------------------------------------
6                        | 2 * 3
46                       | 2 * 23
446                      | 2 * 223
2446                     | 2 * 1223
62446                    | 2 * 31223
762446                   | 2 * 381223
6762446                  | 2 * 3381223
86762446                 | 2 * 43381223
986762446                | 2 * 493381223
4986762446               | 2 * 2493381223
34986762446              | 2 * 17493381223
634986762446             | 2 * 317493381223
9634986762446            | 2 * 4817493381223
59634986762446           | 2 * 29817493381223
959634986762446          | 2 * 479817493381223
9959634986762446         | 2 * 4979817493381223
39959634986762446        | 2 * 19979817493381223
439959634986762446       | 2 * 219979817493381223
8439959634986762446      | 2 * 4219979817493381223
48439959634986762446     | 2 * 24219979817493381223
248439959634986762446    | 2 * 124219979817493381223
4248439959634986762446   | 2 * 2124219979817493381223
84248439959634986762446  | 2 * 42124219979817493381223
984248439959634986762446 | 2 * 492124219979817493381223

The next semiprime is 10 = 2 x 5 stops right where it starts and no digits added its left will produce a semiprime. Next we have 14 = 2 x 7 (permalink):

Semiprime            | Factorisation
---------------------------------------------
14                   | 2 * 7
214                  | 2 * 107
7214                 | 2 * 3607
87214                | 2 * 43607
187214               | 2 * 93607
5187214              | 2 * 2593607
35187214             | 2 * 17593607
735187214            | 2 * 367593607
5735187214           | 2 * 2867593607
95735187214          | 2 * 47867593607
495735187214         | 2 * 247867593607
3495735187214        | 2 * 1747867593607
53495735187214       | 2 * 26747867593607
353495735187214      | 2 * 176747867593607
6353495735187214     | 2 * 3176747867593607
16353495735187214    | 2 * 8176747867593607
316353495735187214   | 2 * 158176747867593607

The next semiprime is 15 = 3 x 5 and it produces the following chain (permalink):

Semiprime            | Factorisation
---------------------------------------------
15                   | 3 * 5
415                  | 5 * 83
7415                 | 5 * 1483
27415                | 5 * 5483
927415               | 5 * 185483
7927415              | 5 * 1585483
97927415             | 5 * 19585483
597927415            | 5 * 119585483
6597927415           | 5 * 1319585483
66597927415          | 5 * 13319585483
366597927415         | 5 * 73319585483
3366597927415        | 5 * 673319585483
33366597927415       | 5 * 6673319585483
733366597927415      | 5 * 146673319585483
9733366597927415     | 5 * 1946673319585483
69733366597927415    | 5 * 13946673319585483
869733366597927415   | 5 * 173946673319585483
9869733366597927415  | 5 * 1973946673319585483
49869733366597927415 | 5 * 9973946673319585483

However, when we input 21 = 3 x 7 the program times out. As Gemini says:

The timeout occurs because some starting numbers, like 21, spawn massive branching paths of valid semiprimes. As the numbers grow larger with each prepended digit, the prime factorization calculations become increasingly computationally expensive.

Setting a maximum depth prevents a timeout but we never get to see the end of the chain. With a maximum depth of 10, we get the following chain of semiprimes:

Semiprime            | Factorisation
---------------------------------------------
21                   | 3 * 7
121                  | 11 * 11
1121                 | 19 * 59
81121                | 23 * 3527
181121               | 71 * 2551
2181121              | 853 * 2557
32181121             | 7 * 4597303
932181121            | 139 * 6706339
3932181121           | 11 * 357471011
33932181121          | 87683 * 386987

By contrast, the semiprime 25 leads to a dead end.

33 = 3 x 11 has the same problem as 21. As I said to Gemini:

The problem with the generation of the semiprime chain seems to be ensuring that it is the longest possible chain. Let’s not try to ensure this. Let’s start with a semiprime like 21 and look for the smallest possible digit that, appended to the left, produces a new semiprime. Let’s proceed on that basis until no suitable digit can be found and the chain ends. Can you construct a chain based on that criterion (with no maximum depth specified). Output as before: table and comma-separated list.

The resultant program produced this output for 21 when it was revised (permalink):

Semiprime                 | Factorisation
---------------------------------------------
21                        | 3 * 7
121                       | 11 * 11
1121                      | 19 * 59
81121                     | 23 * 3527
181121                    | 71 * 2551
2181121                   | 853 * 2557
32181121                  | 7 * 4597303
932181121                 | 139 * 6706339
3932181121                | 11 * 357471011
33932181121               | 87683 * 386987
633932181121              | 181 * 3502387741
2633932181121             | 3 * 877977393707
52633932181121            | 17 * 3096113657713
252633932181121           | 976453 * 258726157
3252633932181121          | 36187 * 89884044883
63252633932181121         | 31 * 2040407546199391
363252633932181121        | 3851 * 94326833012771
3363252633932181121       | 757811 * 4438115353211
53363252633932181121      | 3 * 17787750877977393707
153363252633932181121     | 103 * 1488963617805166807
3153363252633932181121    | 809657423 * 3894688250927
73153363252633932181121   | 13 * 5627181788664148629317
373153363252633932181121  | 331537 * 1125525546930309233
2373153363252633932181121 | 35724754739 * 66428821711739

The downside of course is that the longest possible chain is not uncovered. Here is the result for 14 when the program is now run (permalink).

Semiprime            | Factorisation
---------------------------------------------
14                   | 2 * 7
214                  | 2 * 107
1214                 | 2 * 607
21214                | 2 * 10607
121214               | 2 * 60607

Here is the result for 33 (permalink):

Semiprime            | Factorisation
---------------------------------------------
33                   | 3 * 11
133                  | 7 * 19
1133                 | 11 * 103
21133                | 7 * 3019
121133               | 29 * 4177
2121133              | 7 * 303019
22121133             | 3 * 7373711
122121133            | 107 * 1141319
9122121133           | 4363 * 2090791
39122121133          | 19 * 2059059007
139122121133         | 2801 * 49668733
4139122121133        | 3 * 1379707373711
44139122121133       | 13 * 3395317086241
144139122121133      | 683 * 211038246151
8144139122121133     | 60510661 * 134590153
18144139122121133    | 11 * 1649467192920103

The program works quite well for larger semiprimes too. Take 28293 as an example (permalink):

Semiprime            | Factorisation
---------------------------------------------
28293                | 3 * 9431
428293               | 53 * 8081
1428293              | 131 * 10903
11428293             | 3 * 3809431
211428293            | 9419 * 22447
2211428293           | 6073 * 364141
12211428293          | 4073 * 2998141
312211428293         | 7433 * 42003421
2312211428293        | 13 * 177862417561
32312211428293       | 4967 * 6505377779
332312211428293      | 3659 * 90820500527
4332312211428293     | 18301 * 236725436393
84332312211428293    | 41 * 2056885663693373
284332312211428293   | 3 * 94777437403809431
2284332312211428293  | 17 * 134372488953613429

I'll investigate the generation of right truncatable semiprimes in a future post.

Truncatable Primes Revisited

The following information is true:

The largest left-truncatable prime in base 10 is the 24-digit number:$$357,686,312,646,216,567,629,137$$A left-truncatable prime is a prime number that contains no zeros and remains prime every time you successively strip away its leading (leftmost) digit. 

However, the same information presented visually where it has a lot more impact.

The largest right truncatable number (73939133) is far shorter because there is the constraint that the right-most digit can only be 1, 3, 7 or 9. A similar inverted pyramid will look like this. Note that 7 is the final prime left as was the case with the right truncatable record holder.

73939133

7393913

739391

73939

7393

739

73

7

If we start with the digit 3, the only other possible choice, and build the longest possible chain of left truncatable primes we end up with the following 20 primes (compared to 24 when we started with the digit 7 - permalink):

3

83

883

6883

76883

676883

6676883

36676883

536676883

3536676883

13536676883

213536676883

7213536676883

57213536676883

957213536676883

4957213536676883

84957213536676883

484957213536676883

6484957213536676883

36484957213536676883

Creating a chain of right-truncatable primes gives us eight primes just as it did when starting with the digit 7 (permalink):

3

37

373

3733

37337

373379

3733799

37337999

We could also start with 2 and in that case we get the following right-truncatable primes:

2

29

293

2939

29399

293999

2939999

29399999

Rotate and Add

26947 is a prime, in fact it's a left truncatable prime. This means that with successive removal of digits from the left, the resulting number is still prime. Thus we have 6947, 947, 47 and 7 all being prime. The number was brought to my attention because it represents my diurnal age today: 12th January 2023. Coincidentally if we write this date as 12-1-2023 and then concatenate the digits to form 1212023, this number too is prime.

However, 26947 is also of interest in connection to a so-called rotate and add operation that can be applied to any integer number. If the number has an even number of digits, let's say 1234, then we divide the number into two parts of equal length (12|34) and swap the two parts (34|12) to form 3412. If the number has an odd number of digits, let's say 12345, then we leave the central digit unchanged (12|3|45) but swap the left and right hand parts (45|3|12) to form 45312.

Let's consider primes in the range up to 1000. How many of them will remain prime under this operation. These are the primes that remain prime:

229, 239, 241, 257, 269, 271, 277, 281, 439, 443, 463, 467, 479, 499, 613, 641, 653, 661, 673, 677, 683, 691, 811, 823, 839, 863, 881

Let's take 229 as an example. The operation leads to its rotation (where it becomes 922) and its addition to its rotated form (229 + 922) leads to 1151 which is prime. Primes such as these form OEIS A086002:

A086002

Primes which when added to their own rotation yield a prime.

26947 is one such prime because its rotation (47926) and its addition to this rotation (26947 + 47926) generates the prime 74873. However, when the operation is applied to this new prime, the result is still a prime. This is because 74873 + 73874 = 148747 which is prime. Primes such as this are much rarer and constitute OEIS A086003:

A086003

Primes which remain prime after one and after two applications of the rotate-and-add operation of A086002.

The initial members of this sequence are:

271, 281, 10853, 10903, 10939, 12917, 12919, 16603, 16673, 16823, 16843, 18671, 18911, 18913, 20929, 22817, 22907, 24907, 26813, 26833, 26903, 26947, 28661, 28901, 28921, 30809, 30829, 32831, 32917, 32941, 34939, 36653, 36913, 38651

Unfortunately, if we repeat the process, 26947 does not survive but other numbers do and these constitute OEIS A086004: 

A086004

Primes which remain prime after one and after two and after three applications of the rotate-and-add operation of A086002.

The initial members of this sequence are (permalink):

12917, 12919, 18911, 18913, 22907, 24907, 26903, 28901, 1088063, 1288043, 1408031, 1428029, 1528019, 100083679, 100280419, 100283849, 100483847, 100692793, 100880413, 101080159, 101283839, 101683093, 101683663, 102080149

None of the primes listed above survive another round but there are larger primes that do and these constitute OEIS A261458:

A261458

Primes which remain prime after one, two, three and four applications of the rotate-and-add operation of A086002.

The initial members of this sequence are:

10010905789, 10028905771, 10036905763, 10050905749, 10056905743, 10060905739, 10070905729, 10080905719, 10092905707, 10098905701, 10102905697, 10106905693, 10108905691, 10112905687, 10130905669, 10160905639, 10172905627, 10176905623, 10188905611, 10190905609 

In general, it can be noted that rotation and addition of primes with even numbers of digits never yields a prime. This can be seen, using \(ab\) as an example because the rotation \(ba\) and addition generates \(10a+b+10b+a=11a+11b=11 \times (a+b)\) which is composite.

As far as I know, no primes have been found that survive five applications of the operation and according the the comments to OEIS A261458 six applications can never generate a prime.

The rotate and add operation of course does not need to be confined to primes. For example. we could consider semiprimes that remain semiprimes under one, two, three etc. applications of the operation. Such an investigation could form the basis of a future post.