There's a section in the Numbers Aplenty web page that divides a number into two smaller numbers and then adds the two together to form a new number with some interesting property. For example, the number associated with my diurnal age today is 27876 and Numbers Aplenty has this to say about it:
It can be divided in two parts (27 and 876) that added together give the 42nd triangular number, specifically:
\(27876 \rightarrow 27 + 876 = 903 = \text{T}_{42}\)
Normally I ignore this section of the page as the operation seems a little contrived and this was initially the case yesterday when I was searching for interesting properties associated with 27875. In frustration, I turned to Gemini for help and this is the rather long-winded prompt that I gave it:
I'm trying to find something of mathematical interest about the number 27875. I've looked at the entries for the number in the OEIS (Online Encyclopedia of Integer Sequences) at oeis.org and also the information located at Numbers Aplenty (https://www.numbersaplenty.com/27875). These are my usual sources of information about the number but I'm not satisfied with the information that I found there. I'm looking for some property of the number that is quirky, curious or unique. You'll have to apply some lateral thinking and creativity to come up with such a property. Remember that the property must be mathematical but it can fall into the category of recreational mathematics rather than academic mathematics. The number may have significance or importance in an area outside of mathematics such as physics, geography, demographics etc.
Here was Gemini's response:
A particularly curious property of 27875 emerges from a simple, recreational manipulation of its digits. The number 27875 can be split into two parts: the number formed by the first three digits, 278, and the number formed by the last two digits, 75. When you add these two parts together, you get a special result:
\(278+75=353\)The sum, 353, is a palindromic prime number—it reads the same forwards and backwards, and its only divisors are 1 and itself. palindromic primes are relatively uncommon, and finding one through this kind of "split-sum" operation is a quirky feature.
Diving Deeper 🧐
The property becomes even more unique when we look at the prime number index.
The number 353 is the 71st prime number.
The index itself, 71, is also a prime number.
This makes 353 a prime index prime, adding another layer of mathematical interest to the result derived from 27875.
So Gemini came up with a property of the number that was already on the Numbers Aplenty web page but it did add the additional information that the palindrome was prime and that this prime had a prime index. This got me thinking as to what other five digit numbers had the property that they could be split into two parts and have the two parts add together to form a prime number with a prime index.
I developed an algorithm (permalink) to find such numbers in the range from 27875 to 40000. It turns out that there are 83 such numbers with one of them (29162) able to be split in two different ways:$$ \begin{align} 29162 &\rightarrow 29+162 =191 = \text{P}_{43} \\ 29162 &\rightarrow 291+62 =353 = \text{P}_{71} \end{align}$$Here is a list of the 83 numbers:
27875, 27892, 27974, 28163, 28172, 28271, 28325, 28469, 28568, 28667, 28766, 28769, 28865, 28891, 28964, 29162, 29162, 29261, 29324, 29459, 29558, 29657, 29756, 29768, 29855, 29954, 31142, 31241, 31322, 31439, 31538, 31637, 31736, 31766, 31835, 31888, 31934, 32132, 32159, 32231, 32321, 32429, 32528, 32627, 32726, 32765, 32825, 32887, 32924, 33122, 33158, 33221, 33419, 33518, 33617, 33716, 33764, 33815, 33886, 33914, 34112, 34157, 34211, 34319, 34763, 34885, 35156, 35318, 35762, 35884, 36155, 36317, 36761, 36883, 37154, 37316, 37882, 38153, 38315, 38759, 38881, 39152, 39314, 39758
The details are as follows:
number first second palindrome prime index
27875 278 75 353 71
27892 27 892 919 157
27974 279 74 353 71
28163 28 163 191 43
28172 281 72 353 71
28271 282 71 353 71
28325 28 325 353 71
28469 284 69 353 71
28568 285 68 353 71
28667 286 67 353 71
28766 287 66 353 71
28769 28 769 797 139
28865 288 65 353 71
28891 28 891 919 157
28964 289 64 353 71
29162 29 162 191 43
29162 291 62 353 71
29261 292 61 353 71
29324 29 324 353 71
29459 294 59 353 71
29558 295 58 353 71
29657 296 57 353 71
29756 297 56 353 71
29768 29 768 797 139
29855 298 55 353 71
29954 299 54 353 71
31142 311 42 353 71
31241 312 41 353 71
31322 31 322 353 71
31439 314 39 353 71
31538 315 38 353 71
31637 316 37 353 71
31736 317 36 353 71
31766 31 766 797 139
31835 318 35 353 71
31888 31 888 919 157
31934 319 34 353 71
32132 321 32 353 71
32159 32 159 191 43
32231 322 31 353 71
32321 32 321 353 71
32429 324 29 353 71
32528 325 28 353 71
32627 326 27 353 71
32726 327 26 353 71
32765 32 765 797 139
32825 328 25 353 71
32887 32 887 919 157
32924 329 24 353 71
33122 331 22 353 71
33158 33 158 191 43
33221 332 21 353 71
33419 334 19 353 71
33518 335 18 353 71
33617 336 17 353 71
33716 337 16 353 71
33764 33 764 797 139
33815 338 15 353 71
33886 33 886 919 157
33914 339 14 353 71
34112 341 12 353 71
34157 34 157 191 43
34211 342 11 353 71
34319 34 319 353 71
34763 34 763 797 139
34885 34 885 919 157
35156 35 156 191 43
35318 35 318 353 71
35762 35 762 797 139
35884 35 884 919 157
36155 36 155 191 43
36317 36 317 353 71
36761 36 761 797 139
36883 36 883 919 157
37154 37 154 191 43
37316 37 316 353 71
37882 37 882 919 157
38153 38 153 191 43
38315 38 315 353 71
38759 38 759 797 139
38881 38 881 919 157
39152 39 152 191 43
39314 39 314 353 71
39758 39 758 797 139
29162 occurs 2 times
29162 occurs 2 times
