Tuesday, 29 July 2025

Split and Add

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

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