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.

Right Truncatable Semiprimes

Having learned from the experience of creating left truncable semiprimes in the previous post, I won't try creating the longest possible chain associated with a given semiprime. Instead, starting with a given semiprime, digits 0 to 9 will be added to the right of the semiprime until a semiprime is found and the process repeated until none of the digits from 0 to 9 can create a semiprime. Let's start with the first semiprime 6 = 2 x 3 (permalink):

Semiprime            | Factorisation
---------------------------------------------
6                    | 2 * 3
62                   | 2 * 31
622                  | 2 * 311
6227                 | 13 * 479
62277                | 3 * 20759
622771               | 23 * 27077
6227711              | 7 * 889673
62277118             | 2 * 31138559
622771187            | 1889 * 329683
6227711873           | 65267 * 95419
62277118733          | 137 * 454577509
622771187339         | 7 * 88967312477
6227711873391        | 3 * 2075903957797
62277118733913       | 3 * 20759039577971

Next we have 10 = 2 x 5 (permalink):

Semiprime            | Factorisation
---------------------------------------------
10                   | 2 * 5
106                  | 2 * 53
1067                 | 11 * 97
10671                | 3 * 3557
106717               | 13 * 8209
1067173              | 19 * 56167
10671731             | 7 * 1524533
106717315            | 5 * 21343463
1067173151           | 11 * 97015741
10671731513          | 653 * 16342621
106717315135         | 5 * 21343463027
1067173151351        | 523 * 2040484037

Next we have 14 = 2 x 7 (permalink).

Semiprime            | Factorisation
---------------------------------------------
14                   | 2 * 7
141                  | 3 * 47
1411                 | 17 * 83
14111                | 103 * 137
141119               | 11 * 12829
1411193              | 7 * 201599
14111933             | 11 * 1282903
141119339            | 83 * 1700233

Next we have 15 = 3 x 5 (permalink).

Semiprime            | Factorisation
---------------------------------------------
15                   | 3 * 5
155                  | 5 * 31
1555                 | 5 * 311
15553                | 103 * 151
155531               | 43 * 3617

Now we'll try 21 = 3 x 7 (permalink).

Semiprime            | Factorisation
---------------------------------------------
21                   | 3 * 7
213                  | 3 * 71
2138                 | 2 * 1069
21381                | 3 * 7127
213811               | 13 * 16447
2138111              | 61 * 35051
21381117             | 3 * 7127039

Let's try a larger number like 28293.

Semiprime            | Factorisation
---------------------------------------------
28293                | 3 * 9431
282937               | 31 * 9127
2829371              | 1571 * 1801
28293711             | 3 * 9431237
282937111            | 15959 * 17729
2829371119           | 331 * 8547949

Clearly the chains are shorter when creating right truncatable semiprimes.

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

Supercharged Semiprimes

 There is a set of non-square semiprimes defined by two criteria:

• the two prime factors concatenate in either order to form two new semiprimes
• the digits of the two prime factors can each be rearranged to form semiprimes

28291 is an example of such a semiprime. Let's examine its properties where || represents concatenation:$$ \begin{align} 28291 &=19 \times 1489\\ 19 \, || \, 1489 &= 191489 \\ &= 53 \times 3613 \\ 1489 \, || \,19  &=148919 \\ &= 137 \times 1087 \\ 19 &\rightarrow 91 \\&=7 \times 13 \\1489 &\rightarrow 1894 \\ &= 2 \times 947 \end{align}$$Up to 40000, the semiprimes that meet both criteria are (permalink):

779, 817, 1121, 1763, 1957, 2071, 2869, 3173, 3403, 3629, 4579, 4687, 4897, 5149, 5977, 6973, 7009, 7181, 7261, 7367, 7493, 8077, 8341, 8413, 8549, 8851, 8977, 9071, 9937, 10123, 10363, 10393, 10679, 10697, 10951, 11051, 11419, 11647, 12407, 12599, 13207, 14317, 14473, 14719, 14809, 14953, 15007, 15143, 15529, 15637, 15751, 16283, 16297, 16321, 16351, 16789, 16873, 17671, 18391, 18563, 18643, 19177, 19247, 19451, 20081, 20311, 20653, 20729, 20989, 21071, 21223, 21733, 21829, 21887, 21971, 22313, 22361, 22489, 22819, 22837, 22879, 22987, 23083, 23351, 23521, 24257, 24377, 24559, 24751, 24757, 24823, 25061, 25843, 25901, 26179, 26239, 26617, 26671, 26969, 27089, 27161, 27199, 27331, 27383, 27589, 27913, 28177, 28291, 28801, 28907, 28937, 28999, 29149, 29329, 29621, 30301, 30571, 30847, 31111, 31921, 32101, 32239, 32293, 32387, 32651, 32699, 33307, 33907, 33991, 34093, 34571, 34579, 34633, 34873, 34927, 35137, 35209, 35341, 35389, 35587, 35701, 35881, 36031, 36199, 36689, 37069, 37127, 37867, 37901, 38021, 38141, 38173, 38323, 38477, 38497, 38989, 39187, 39433, 39707, 39757

Fermat Numbers

A Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a positive integer of the form: $$ \text{F}_n=2^{2^n}+1 $$ where n is a non-negative integer. The first few Fermat numbers are:

• \(\text{F}_0=3\)
• \(\text{F}_1=5\)
• \(\text{F}_2=17 \)
• \(\text{F}_3=257\)
• \(\text{F}_4=65537\)
• \(\text{F}_5=4294967297\)
• \(\text{F}_6=18446744073709551617\)
• \(\text{F}_7=340282366920938463463374607431768211457\)

The numbers form the initial members of sequence A000215 in the OEIS. The only known Fermat primes are the first five members of this sequence (3, 5, 17, 257 and 65537).

The number associated with my diurnal age today, \( \textbf{28289}\), is a prime and oddly it has a connection to these Fermat numbers via its membership in OEIS A229856:

   A229856: primes of the form \(384 \times k + 257\) where \(k \geq 0\)

The initial members are:

257, 641, 1409, 3329, 4481, 7937, 9473, 9857, 11393, 11777, 12161, 13313, 13697, 14081, 15233, 16001, 17921, 19073, 19457, 19841, 21377, 23297, 25601, 28289, 30593, 30977, 35201, 35969, 36353, 37889, 38273, 39041

It's stated in the comments to this OEIS sequence that:

Every Fermat number greater than 257 has a prime factor of the form \(384 \times k + 257, k > 0\).

Let's test this out with the initial Fermat numbers greater than 257. Now \( \text{F}_4=65537\) is a prime and so it is its own prime factor and we find that:$$65537 = 384 \times 170 + 257$$The next Fermat numbers listed above are all composite so we need to check their prime factors:$$ \begin{align} \text{F}_5=4294967297 &= 641 \times 6700417 \\ 641 &= 384 \times 1 + 257 \end{align}$$ $$ \begin{align} \text{F}_6=18446744073709,551,617 &=274177 × 67280421310721\\ 67280421310721 &= 384 \times 175209430496+257 \end{align}$$I won't attempt \( \text{F}_7 \) as it's too lengthy but the point is made. There's an explanation that Gemini provided as to why at least one prime factor of a Fermat number must have this form. You can check that out here.

Variations on Magnanimous Numbers

Today I'm 28285 days old and one of the properties of this number is that it's magnanimous, meaning that inserting a "+" between any two digits produces a prime. In this case we have:

• 2+8285 = 8287 is prime
  
  
• 28+285 = 313 is prime
  
  
• 282+85 = 367 is prime
  
  
• 2828+5 = 2833 is prime

I review these sorts of numbers in my post titled Magnanimous Numbers from December of 2020. This got me thinking about subtraction instead of addition or to put it formally and applying the absolute value operator:

| Minuend - Subtrahend | = | Difference |

Take as an example, the number 28041 where we have:

• | 2 - 8041 | = 8039 is prime
  
  
• | 28 - 41 | = 13 is prime
  
  
• | 280 - 41 | = 239 is prime
  
  
• | 2804 - 1 | = 2803 is prime

Here is the full list of 230 such numbers up to 40000 (permalink):

13, 14, 16, 18, 20, 24, 25, 27, 29, 30, 31, 35, 36, 38, 41, 42, 46, 47, 49, 50, 52, 53, 57, 58, 61, 63, 64, 68, 69, 70, 72, 74, 75, 79, 81, 83, 85, 86, 92, 94, 96, 97, 103, 108, 114, 118, 130, 132, 138, 154, 174, 190, 198, 207, 209, 225, 245, 263, 269, 285, 301, 310, 332, 334, 356, 370, 376, 392, 409, 421, 441, 447, 463, 465, 487, 503, 507, 518, 536, 552, 558, 601, 623, 629, 643, 665, 667, 689, 709, 710, 714, 730, 736, 754, 774, 790, 796, 801, 845, 867, 907, 912, 956, 970, 992, 1003, 1114, 1152, 1158, 1332, 1354, 1510, 1614, 1758, 1930, 1992, 1998, 2007, 2009, 2025, 2241, 2421, 2465, 2603, 2609, 2663, 2685, 2729, 2825, 2841, 2865, 3154, 3310, 3370, 3376, 3392, 3436, 3512, 3596, 3634, 3730, 3790, 3956, 3970, 4021, 4047, 4087, 4201, 4245, 4341, 4401, 4467, 4603, 4663, 4665, 4687, 4801, 4867, 5052, 5156, 5514, 5592, 5658, 5952, 5996, 6001, 6023, 6029, 6043, 6065, 6067, 6089, 6163, 6203, 6245, 6269, 6365, 6563, 6607, 6623, 6689, 6863, 7009, 7134, 7270, 7314, 7390, 7396, 7570, 7734, 7936, 8069, 8241, 8265, 8285, 8607, 8625, 8667, 8669, 8847, 8865, 8885, 9092, 9290, 9356, 9736, 9896, 9976, 9992, 10003, 11532, 11598, 11952, 15154, 17574, 20007, 20063, 20841, 22041, 22401, 24029, 24203, 24263, 24623, 26009, 26265, 27029, 28041, 33190, 37396

With multiplication we would need to add 1 or subtract 1. Let's take 28034 as an example of the former where we have:

• 2 x 8034 + 1 = 16069 is prime
  
  
• 28 x 34 + 1 = 953 is prime
  
  
• 280 x 34 + 1 = 9521 is prime
  
  
• 2803 x 4 + 1 = 11213 is prime

The 421 numbers that satisfy in the range up to 40000 are (permalink):

11, 12, 14, 16, 21, 22, 23, 25, 26, 28, 29, 32, 34, 36, 41, 43, 44, 47, 49, 52, 56, 58, 61, 62, 63, 65, 66, 67, 74, 76, 82, 85, 89, 92, 94, 98, 101, 104, 106, 112, 116, 118, 128, 136, 142, 152, 166, 178, 182, 202, 203, 205, 209, 221, 223, 229, 236, 244, 248, 254, 256, 263, 265, 274, 281, 298, 302, 306, 326, 332, 336, 352, 374, 376, 392, 394, 401, 407, 425, 428, 434, 448, 449, 484, 487, 502, 508, 512, 526, 542, 548, 556, 562, 566, 584, 586, 601, 603, 607, 616, 625, 626, 632, 647, 652, 658, 661, 663, 666, 704, 706, 718, 728, 734, 748, 766, 778, 794, 805, 812, 829, 832, 844, 865, 874, 884, 902, 934, 958, 968, 982, 992, 1001, 1004, 1006, 1018, 1052, 1106, 1108, 1156, 1162, 1178, 1192, 1226, 1228, 1256, 1312, 1352, 1448, 1498, 1568, 1616, 1658, 1708, 1726, 1756, 1862, 1886, 1982, 2002, 2003, 2009, 2023, 2026, 2065, 2074, 2086, 2176, 2204, 2216, 2221, 2228, 2243, 2281, 2306, 2336, 2384, 2386, 2443, 2524, 2576, 2645, 2648, 2686, 2716, 2774, 2776, 2849, 2876, 2998, 3002, 3006, 3034, 3074, 3136, 3152, 3224, 3244, 3302, 3404, 3412, 3526, 3542, 3566, 3574, 3592, 3704, 3734, 3932, 4001, 4007, 4034, 4043, 4078, 4102, 4142, 4168, 4234, 4265, 4274, 4304, 4402, 4423, 4447, 4504, 4685, 4724, 4807, 4825, 4918, 4982, 5008, 5036, 5066, 5162, 5198, 5206, 5302, 5344, 5366, 5506, 5518, 5534, 5542, 5558, 5594, 5608, 5612, 5708, 5726, 5786, 5918, 6001, 6003, 6007, 6056, 6076, 6112, 6166, 6178, 6182, 6245, 6263, 6268, 6382, 6412, 6443, 6452, 6467, 6518, 6536, 6601, 6605, 6607, 6632, 6667, 6766, 6802, 6902, 7004, 7006, 7094, 7118, 7126, 7244, 7384, 7424, 7516, 7606, 7706, 7886, 7918, 7948, 8005, 8042, 8056, 8186, 8201, 8249, 8261, 8266, 8324, 8332, 8462, 8474, 8516, 8536, 8542, 8602, 8662, 8794, 8912, 8942, 8984, 9002, 9034, 9112, 9128, 9158, 9254, 9278, 9428, 9494, 9502, 9532, 9652, 9704, 9748, 9932, 9992, 10004, 10018, 10028, 10136, 10312, 10336, 10448, 11102, 11276, 11482, 11552, 11578, 11608, 11662, 12026, 12206, 12656, 12728, 13256, 13312, 14002, 14098, 14158, 14548, 14758, 14798, 14812, 14968, 15208, 15856, 15952, 16142, 17126, 18116, 18536, 18736, 18962, 19012, 19402, 19768, 19832, 20002, 20021, 20254, 20849, 20866, 21778, 22178, 22616, 22778, 22948, 23834, 23876, 24043, 24118, 24334, 24424, 24824, 25228, 25298, 25366, 25556, 25844, 26005, 26516, 26665, 27028, 27356, 27784, 28034, 28516, 28874, 29708, 30274, 30362, 30662, 30704, 32024, 32066, 32234, 33106, 33182, 33784, 34492, 35206, 36676, 38032, 38362, 38512, 38734, 39532

If instead we subtract 1, we get the following 400 numbers in the range up to 40000 (permalink):

13, 14, 16, 18, 22, 23, 24, 26, 27, 29, 31, 32, 34, 36, 38, 41, 42, 43, 45, 46, 48, 54, 56, 61, 62, 63, 64, 65, 67, 68, 69, 72, 76, 81, 83, 84, 86, 89, 92, 96, 98, 103, 106, 108, 114, 124, 138, 154, 162, 168, 174, 184, 198, 203, 204, 207, 209, 212, 222, 236, 242, 264, 269, 296, 301, 302, 306, 308, 324, 334, 336, 338, 384, 386, 398, 402, 405, 406, 421, 426, 427, 441, 456, 463, 468, 492, 496, 504, 512, 522, 536, 548, 572, 588, 596, 601, 603, 604, 607, 608, 618, 629, 638, 642, 643, 664, 667, 684, 702, 706, 714, 726, 762, 784, 786, 792, 801, 803, 806, 809, 825, 834, 845, 846, 863, 876, 902, 908, 912, 948, 956, 962, 972, 986, 992, 996, 1006, 1038, 1062, 1084, 1104, 1152, 1308, 1314, 1368, 1422, 1504, 1524, 1548, 1608, 1614, 1654, 1662, 1692, 1734, 1752, 1824, 1854, 1864, 1884, 1968, 1992, 2003, 2007, 2012, 2036, 2052, 2064, 2112, 2274, 2427, 2465, 2486, 2609, 2612, 2664, 2724, 2736, 2805, 3002, 3008, 3086, 3198, 3234, 3304, 3318, 3368, 3408, 3498, 3504, 3596, 3624, 3634, 3638, 3668, 3786, 3926, 3976, 4005, 4006, 4021, 4026, 4083, 4152, 4158, 4188, 4207, 4221, 4246, 4272, 4306, 4326, 4396, 4407, 4458, 4692, 4801, 4818, 4843, 4906, 4962, 4992, 5004, 5102, 5148, 5168, 5172, 5202, 5214, 5334, 5376, 5418, 5462, 5492, 5538, 5604, 5622, 5756, 5784, 6001, 6004, 6008, 6012, 6043, 6045, 6067, 6098, 6264, 6308, 6334, 6368, 6402, 6465, 6542, 6603, 6665, 6714, 6754, 6774, 6834, 6858, 6912, 6998, 7002, 7152, 7242, 7422, 7452, 7476, 7546, 7554, 7564, 7806, 7876, 7896, 7926, 7996, 8003, 8006, 8063, 8154, 8184, 8198, 8289, 8445, 8483, 8598, 8685, 8784, 8948, 9296, 9542, 9672, 9756, 9806, 9836, 9902, 9972, 9976, 10014, 10024, 10062, 10234, 10422, 10548, 10662, 11088, 11118, 11202, 12004, 12504, 12522, 12634, 13308, 13638, 14022, 14448, 14592, 14958, 15114, 15438, 15504, 15648, 15684, 16008, 16038, 16654, 16962, 17742, 18054, 18468, 19398, 20007, 20012, 20112, 20154, 20205, 20289, 20427, 20724, 20784, 21102, 21212, 21272, 21774, 21792, 21962, 22014, 22122, 22164, 22212, 22242, 22401, 22605, 22821, 22824, 22962, 23486, 24086, 24602, 25122, 25386, 26004, 26412, 26574, 26669, 27036, 27252, 27542, 27756, 27912, 28056, 28284, 28896, 29072, 29292, 29402, 29702, 30086, 30098, 30604, 30848, 31234, 32004, 32514, 33336, 33354, 33368, 33726, 34908, 35706, 36124, 36168, 37134, 37204, 37356, 38036, 38184, 39006, 39048

Let's take the last member, 39048, as an example:

• 3 x 9048 - 1 = 27143 is prime
  
  
• 39 x 48 - 1 = 1871 is prime
  
  
• 390 x 48 - 1 = 18719 is prime
  
  
• 3904 x 8 - 1 = 31231 is prime

Semiprime Partitions of Semiprimes

The number associated with my diurnal age today, \( \textbf{28284} \), has a property that grants it membership in OEIS A338902:

A338902: number of integer partitions of the n-th semiprime into semiprimes.

I got Gemini to write a program to find the number of suitable partitions for integers from 1 to 50 displaying the results as both a table and a list (permalink):

Semiprime | Partitions
-----------------------
4         | 1
6         | 1
9         | 1
10        | 2
14        | 3
15        | 2
21        | 4
22        | 7
25        | 7
26        | 10
33        | 17
34        | 25
35        | 21
38        | 34
39        | 34
46        | 73
49        | 87
51        | 103
55        | 149
57        | 176
58        | 206
62        | 281
65        | 344
69        | 479
74        | 725
77        | 881
82        | 1311
85        | 1597
86        | 1742
87        | 1841
91        | 2445
93        | 2808
94        | 3052
95        | 3222
106       | 6784
111       | 9298
115       | 11989
118       | 14533
119       | 15384
121       | 17414
122       | 18581
123       | 19680
129       | 28284
133       | 35862
134       | 38125
141       | 57095
142       | 60582
143       | 64010
145       | 71730
146       | 76016

1, 1, 1, 2, 3, 2, 4, 7, 7, 10, 17, 25, 21, 34, 34, 73, 87, 103, 149, 176, 206, 281, 344, 479, 725, 881, 1311, 1597, 1742, 1841, 2445, 2808, 3052, 3222, 6784, 9298, 11989, 14533, 15384, 17414, 18581, 19680, 28284, 35862, 38125, 57095, 60582, 64010, 71730, 76016

As can be seen, the integer \( \textbf{43} \) has \( \textbf{28284} \) partitions compared to its 63261 unrestricted partitions.

Similar results could be created for primes and triprimes. Here are the result for the primes (permalink). The numbers form OEIS A056768.

Prime   | Partitions
---------------------
2       | 1
3       | 1
5       | 2
7       | 3
11      | 6
13      | 9
17      | 17
19      | 23
23      | 40
29      | 87
31      | 111
37      | 219
41      | 336
43      | 413
47      | 614
53      | 1083
59      | 1850
61      | 2198
67      | 3630
71      | 5007
73      | 5861
79      | 9282
83      | 12488
89      | 19232
97      | 33439
101     | 43709
103     | 49871
107     | 64671
109     | 73506
113     | 94625
127     | 221265
131     | 279516
137     | 394170
139     | 441250
149     | 766262
151     | 853692
157     | 1175344
163     | 1608014
167     | 1975108
173     | 2675925
179     | 3605666
181     | 3977861
191     | 6447003
193     | 7089299
197     | 8559069
199     | 9397474
211     | 16298212
223     | 27810910
227     | 33121140
229     | 36123177

1, 1, 2, 3, 6, 9, 17, 23, 40, 87, 111, 219, 336, 413, 614, 1083, 1850, 2198, 3630, 5007, 5861, 9282, 12488, 19232, 33439, 43709, 49871, 64671, 73506, 94625, 221265, 279516, 394170, 441250, 766262, 853692, 1175344, 1608014, 1975108, 2675925, 3605666, 3977861, 6447003, 7089299, 8559069, 9397474, 16298212, 27810910, 33121140, 36123177

Here are the result for the triprimes (permalink). 

Triprime  | Partitions
-----------------------
8         | 1
12        | 1
18        | 1
20        | 2
27        | 1
28        | 3
30        | 2
42        | 4
44        | 7
45        | 2
50        | 7
52        | 10
63        | 7
66        | 18
68        | 25
70        | 22
75        | 13
76        | 34
78        | 36
92        | 78
98        | 97
99        | 47
102       | 117
105       | 60
110       | 170
114       | 202
116       | 230
117       | 111
124       | 316
125       | 165
130       | 401
138       | 579
147       | 478
148       | 866
153       | 616
154       | 1086
164       | 1652
165       | 1040
170       | 2076
171       | 1350
172       | 2227
174       | 2418
175       | 1566
182       | 3271
186       | 3802
188       | 4115
190       | 4372
195       | 3587
207       | 5708
212       | 9783

1, 1, 1, 2, 1, 3, 2, 4, 7, 2, 7, 10, 7, 18, 25, 22, 13, 34, 36, 78, 97, 47, 117, 60, 170, 202, 230, 111, 316, 165, 401, 579, 478, 866, 616, 1086, 1652, 1040, 2076, 1350, 2227, 2418, 1566, 3271, 3802, 4115, 4372, 3587, 5708, 9783, 13878, 18291, 13983, 22465, 23913, 27489, 29296, 23092, 31441, 32846 (these are first 60 terms).

These results are base-independent and in general what is being done here is to take an integer with a certain property (primeness for example) and express it as a sum of smaller numbers with the same property. The numbere of ways in which this can be done is being counted. Here are some examples:

• 129 = 4 + 4 + 4 + 6 + 111 ... there are 28284 ways in which this can be done
• 97 = 3 + 5 + 89 ... there are 33439 ways in which this can be done
• 212 = 8 + 12 + 18 + 174 ... there are 9783 ways in which this can be done

Rhonda Numbers Revisited

I first reviewed Rhonda Numbers in an eponymous post on the 16th October 2018. Read that post to find out what defines such numbers. Today, upon turning 28280 days old, I was reminded of them again because this number is a member of OEIS A255731:

A255731: Rhonda numbers in sexagesimal number system.

The program that I wrote back then only covered bases from 2 to 36 so I got Gemini to write a new SageMath program that would accommodate every base. Here is what is generated as output when asked to find all Rhonda numbers in base 60 in the range up to one million (permalink):

3348, 3510, 6750, 17430, 18750, 18876, 18944, 19475, 20564, 21312, 26550, 28280, 37230, 38396, 43940, 48042, 77770, 88270, 91224, 97470, 108882, 111403, 120046, 123630, 181996, 182646, 235467, 253460, 260429, 264735, 278675, 289161, 295960, 296055, 306642, 324394, 325593, 337040, 348641, 361221, 377130, 378444, 398274, 411342, 412930, 441048, 444405, 450528, 453470, 458136, 469098, 499533, 503310, 517803, 533731, 534795, 541807, 547515, 566754, 598695, 612374, 612870, 626535, 630410, 656370, 656750, 667491, 670548, 684456, 701765, 703304, 705256, 706275, 709475, 720279, 750225, 757576, 762745, 765245, 809107, 812658, 821106, 877300, 880045, 881454, 915348, 927303, 929830, 930304, 936573, 936675, 967509, 972196, 973549, 984485, 986895, 998430

I also realised that I hadn't included the identification of Rhonda numbers in my daily number analysis and that deficiency has now been remedied. Here the output of the program that I got Gemini to write for the input 29280 (permalink):

28280 : determination of whether it is a Rhonda number in a given base

--- Evaluating 28280 in Base 60 ---

1. Base-60 Digits: [7, 51, 20]

2. Digit Product: 7 * 51 * 20 = 7140

3. Prime Factorization: 2^3 * 5 * 7 * 101

4. Sum of Factors: 2 + 2 + 2 + 5 + 7 + 101 = 119

5. Target Equation (Base * Sum): 60 * 119 = 7140

RESULT: True. 28280 IS a Rhonda number in base 60.

I've set the range of bases to be investigate to 100,000 so that even a number like 889200 with is a Rhonda number in TEN different bases has an accurate output:

--- Evaluating 889200 in Base 1512 ---

1. Base-1512 Digits: [588, 144]

2. Digit Product: 588 * 144 = 84672

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 1512 * 56 = 84672

RESULT: True. 889200 IS a Rhonda number in base 1512.

-----------------------------------

--- Evaluating 889200 in Base 2760 ---

1. Base-2760 Digits: [322, 480]

2. Digit Product: 322 * 480 = 154560

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 2760 * 56 = 154560

RESULT: True. 889200 IS a Rhonda number in base 2760.

-----------------------------------

--- Evaluating 889200 in Base 5160 ---

1. Base-5160 Digits: [172, 1680]

2. Digit Product: 172 * 1680 = 288960

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 5160 * 56 = 288960

RESULT: True. 889200 IS a Rhonda number in base 5160.

-----------------------------------

--- Evaluating 889200 in Base 7904 ---

1. Base-7904 Digits: [112, 3952]

2. Digit Product: 112 * 3952 = 442624

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 7904 * 56 = 442624

RESULT: True. 889200 IS a Rhonda number in base 7904.

-----------------------------------

--- Evaluating 889200 in Base 9400 ---

1. Base-9400 Digits: [94, 5600]

2. Digit Product: 94 * 5600 = 526400

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 9400 * 56 = 526400

RESULT: True. 889200 IS a Rhonda number in base 9400.

-----------------------------------

--- Evaluating 889200 in Base 10032 ---

1. Base-10032 Digits: [88, 6384]

2. Digit Product: 88 * 6384 = 561792

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 10032 * 56 = 561792

RESULT: True. 889200 IS a Rhonda number in base 10032.

-----------------------------------

--- Evaluating 889200 in Base 11440 ---

1. Base-11440 Digits: [77, 8320]

2. Digit Product: 77 * 8320 = 640640

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 11440 * 56 = 640640

RESULT: True. 889200 IS a Rhonda number in base 11440.

-----------------------------------

--- Evaluating 889200 in Base 12920 ---

1. Base-12920 Digits: [68, 10640]

2. Digit Product: 68 * 10640 = 723520

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 12920 * 56 = 723520

RESULT: True. 889200 IS a Rhonda number in base 12920.

-----------------------------------

--- Evaluating 889200 in Base 14136 ---

1. Base-14136 Digits: [62, 12768]

2. Digit Product: 62 * 12768 = 791616

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 14136 * 56 = 791616

RESULT: True. 889200 IS a Rhonda number in base 14136.

-----------------------------------

--- Evaluating 889200 in Base 15080 ---

1. Base-15080 Digits: [58, 14560]

2. Digit Product: 58 * 14560 = 844480

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 15080 * 56 = 844480

RESULT: True. 889200 IS a Rhonda number in base 15080.

-----------------------------------

28277: A Prime To Be Proud Of

Until today my diurnal age has suffered a drought of prime numbers. Prior to today (3rd September 2026) when I turned 28277 days old, the last prime (28229) occurred on July 17th. This marked a gap of 48 days between successive primes. This is not a record gap but it is impressive. Figure 1 shows the successive record gaps between primes.

Figure 1

I'll enumerate some of 28277's most interesting properties:

PROPERTY 1:

It forms a twin prime with 28279 but it also marks the beginning of gaps of 2, 4, 6, 8 and 10 between successive primes. The progression of primes is thus 28277, 28279, 28283, 28289, 28297, 28307. Such an occurrence is not common and membership is restricted to only three numbers (13901, 21557, 28277) in the range up to 40000. These and subsequent numbers constitute OEIS A190817.

PROPERTY 2

28277 is what is called a "good" prime and I posted about this type of prime in my blog post titled The Good Prime on the 12th of March 2025. As I explained there:

A prime \(p_n\) is said to be \( \textbf{good} \) if \(p_n^{^\textbf{2}}>p_{n-i } \cdot p_{n+i} \) for all \( 1 \leq i < n \).

The good primes from 28277 to 40000 are: 

28277, 28387, 28403, 28493, 28537, 28571, 28591, 28597, 29833, 29983, 30011, 30059, 30089, 30491, 30631, 30637, 30671, 30757, 30803, 31121, 31139, 31147, 31957, 32027, 32051, 32057, 32297, 32969, 33287, 33311, 33329, 34123, 35729, 35747, 35797, 35801, 35831, 35951, 35963, 36433, 36451, 36467, 36523, 36527, 36671, 38113, 38149, 38167, 38177, 38543, 38557, 38593, 38651, 38669, 39079, 39089

PROPERTY 3

28277 has what might be called an "internal prime". Strip away the first and last digits and what remains is 827, a prime number. I discuss these types of numbers in my post titled Numbers Within Numbers from the 21st June 2026. Primes with this property form OEIS A069686:

 A069686: primes whose internal digits form a prime.

From 28277 to 40000, the members of the sequence are:

28277, 28279, 28297, 28393, 28537, 28571, 28573, 28579, 28591, 28597, 28631, 28771, 28813, 28817, 28837, 28871, 28879, 29077, 29191, 29297, 29411, 29473, 29531, 29537, 29671, 29717, 29833, 29837, 29917, 30029, 30059, 30071, 30113, 30119, 30133, 30137, 30139, 30197, 30293, 30313, 30319, 30431, 30539, 30593, 30671, 30677, 30713, 30839, 30893, 30971, 30977, 31013, 31019, 31033, 31039, 31079, 31091, 31139, 31271, 31277, 31319, 31379, 31391, 31393, 31397, 31511, 31513, 31517, 31573, 31793, 31799, 31817, 31973, 31991, 32117, 32119, 32233, 32237, 32297, 32299, 32411, 32413, 32573, 32579, 32633, 32693, 32713, 32717, 32719, 32771, 32779, 32831, 32833, 32839, 32933, 32939, 33071, 33073, 33113, 33119, 33179, 33311, 33317, 33377, 33479, 33493, 33533, 33599, 33679, 33739, 33791, 33797, 33893, 34019, 34211, 34213, 34217, 34313, 34319, 34337, 34439, 34499, 34613, 34631, 34673, 34679, 34871, 34877, 34913, 34919, 35099, 35419, 35573, 35771, 35879, 35933, 35993, 35999, 36011, 36013, 36017, 36073, 36131, 36137, 36191, 36313, 36319, 36433, 36473, 36479, 36599, 36739, 36779, 36833, 36913, 36919, 37013, 37019, 37097, 37199, 37273, 37277, 37337, 37339, 37397, 37511, 37517, 37571, 37573, 37579, 37619, 37691, 37693, 37699, 37871, 37879, 38113, 38119, 38219, 38231, 38237, 38239, 38273, 38299, 38393, 38593, 38639, 38833, 38839, 38873, 39079, 39113, 39119, 39191, 39199, 39293, 39371, 39373, 39419, 39671, 39679, 39719, 39779, 39839, 39971, 39979

PROPERTY 4

28277 gives prime 31931717 when digits become indices of prime numbers. Here we have:

• \(2 \rightarrow p_2=3\)
• \(8 \rightarrow p_8=19\)
• \(7 \rightarrow p_7=17\)

The primes of this sort from 28277 to 40000 are (permalink):

28277, 28289, 28319, 28429, 28807, 28979, 29137, 29347, 29399, 29717, 29819, 29837, 29917, 30059, 30089, 30169, 30187, 30367, 30467, 30469, 30497, 30509, 30689, 30697, 30707, 30727, 31139, 31159, 31177, 31247, 31277, 31337, 31469, 31489, 31687, 31847, 31849, 32099, 32309, 32569, 32717, 32957, 32987, 33049, 33469, 33577, 33629, 33797, 33809, 33889, 33937, 34019, 34129, 34259, 34297, 34327, 34439, 34469, 34607, 34649, 34807, 34819, 35027, 35159, 35407, 35597, 36037, 36107, 36109, 36467, 36469, 36587, 36637, 36809, 36857, 37087, 37139, 37199, 37277, 37337, 37579, 37889, 38189, 38299, 38327, 38459, 38609, 38639, 38699, 38747, 39047, 39119, 39239, 39397, 39667, 39779

PROPERTY 5

28277 is the average of a prime and its emirp in two different ways. The two ways are:

1. \( \dfrac{18773 + 37781}{2}= 28277\)
2. \( \dfrac{19763 + 36791}{2} = 28277\)

I discuss these sorts of primes in my blog post Prime Emirp Pair Averages of 12th May 2023. Such primes form OEIS A178587:

A178587

Primes that are the average of the members of more than one emirp pair.

These primes are few and far between with the initial members being:

14741, 22727, 23327, 24547, 25447, 27067, 28277, 42929, 63541, 65761, 85453, 1217171, 1221221, 1227271, 1243421, 1245421, 1246471, 1250521, 1253521, 1257521, 1261571, 1271671, 1283771, 1327231, 1335331, 1338331, 1339381 

PROPERTY 6

28277 has the following trajectory under the Primes(+) and Non_Primes(-) algorithm: 

\(28277 \rightarrow 28287 \rightarrow 28282 \rightarrow 28272 \rightarrow 28277\)

It is thus a vortical and part of the vortex beginning and ending with 28277. This vortex has 36 captives.

Let Us Count the Ways

I was curious as to how to count the number of ways that a given number can be represented in Loeschian form based on its prime factorisation. I knew there was such a method for counting how many ways a number can be represented as a sum of two squares. I got Gemini to investigate, compare and summarise the results. Notice the use of the ceiling function when determining the number of essentially distinct pairs. Remember also that the invalid primes become valid if raised to an even power but they don't contribute to the count. The \(b_{\textit{i}}\) in the table refers to the exponents of the generating primes.

The algebraic frameworks for expressing integers as \(x^2 + xy + y^2\) and \(x^2 + y^2\) mirror each other directly through their respective complex integer rings.

Characteristic	Loeschian Form \(x^2 + xy + y^2\)	Sum of Two Squares \(x^2 + y^2\)
Complex Ring	Eisenstein Integers \(\mathbb{Z}[e^{2\pi i/3}]\)	Gaussian Integers \(\mathbb{Z}[i]\)
Invalid Primes (Require Even Exponents)	\(p \equiv 2 \pmod 3\)	\(p \equiv 3 \pmod 4\)
Generating Primes (Exponents \(b_i\))	\(p \equiv 1 \pmod 3\)	\(p \equiv 1 \pmod 4\)
Neutral Prime	\(p = 3\)	\(p = 2\)
Unit Symmetries (Multiplier)	6	4
Total Ordered Pairs \((x,y)\)	\(6 \prod (b_i + 1)\)	\(4 \prod (b_i + 1)\)
Essentially Distinct Pairs \(\{x,y\}\)	\(\lceil \frac{1}{2} \prod (b_i + 1) \rceil\)	\(\lceil \frac{1}{2} \prod (b_i + 1) \rceil\)

Factorization Conditions

Both systems demand that primes which cannot be represented natively by the quadratic form must be "squared away" with even exponents in the prime factorization. For Gaussian integers, primes congruent to \(3 \pmod 4\) (such as 3, 7, 11) cannot form sums of two squares. For Eisenstein integers, primes congruent to \(2 \pmod 3\) (such as 2, 5, 11) cannot be Loeschian. If any of these restricted primes have an odd exponent, the integer cannot be represented in that form, and the total count is zero.

Symmetry and Distinct Ways

The difference in the total number of ordered coordinate pairs stems entirely from the geometry of the complex rings. Gaussian integers form a square lattice with 4 units (\(\pm 1, \pm i\)), leading to a rotational symmetry multiplier of 4. Eisenstein integers form a hexagonal lattice with 6 units (\(\pm 1, \pm e^{2\pi i/3}, \pm e^{4\pi i/3}\)), leading to a multiplier of 6.

Once these geometric symmetries are factored out by dividing by the respective unit count, both systems calculate the number of essentially distinct, non-negative integer pairs using the exact same ceiling function formula. The generating primes dictate this final count, while the unique neutral prime in each system scales the base integer without creating new algebraic variations.