A Range Limit of 40000: Why?

In many of my posts when I'm considering sequences, I only look at sequence members whose values do not exceed 40000. Why? The answer to this question relates to the reason that I set this blog up in the first place. It was designed primarily to post about interesting sequences associated with the numbers marking my diurnal age.

If we divide 40000 by 365.2425 (the average number of days in a year) we get slightly more than 109.5 years and there are not many people who live to that ripe old age. Here are milestones, one might say, on the road to oblivion.

Quite a few people won't see 30000 days let alone 40000 but many will and hence the realistic upper limit to the numbers that I normally investigate. The focus of my blog posts is generally the number associated with my diurnal age and the sequences that it can be connected to. For example, today I am \( \textbf{27920} \) days old.

This number found its way into a sequence that I created that involves gapful numbers with the property that not only does the number formed by the concatenation of the first and last digit divide the number but this concatenated number is also the sum of the number's digits. Thus we have:$$ \begin{align} \frac{27920}{20} &=1396 \\ \\ 2 + 7 + 9 + 2 + 0 &= 20 \end{align}$$I described this sequence in a post titled Gapgul Numbers in December of 2024. Interestingly, 27920 also has the property that it has 20 divisors. Only 19 numbers satisfy this additional criterion in the range up to 40000. The conditions to be met are:

• the number is gapful meaning the number formed by concantenating the first and last digits divides the number without remainder
• the sum of the number's digits equals the number formed by the concatenated first and last digits
• the number of divisors of the number equals its sum of digits (and the concatenated number)

Here are the numbers:

1548, 1812, 1908, 10188, 10548, 11268, 12252, 12612, 12708, 13428, 14052, 14412, 15138, 18108, 21984, 26480, 27920, 29360, 39996

Here are the details:

  number        factor          concat   dividend   S0D   divisors

  1548     2^2 * 3^2 * 43         18       86         18    18
  1812     2^2 * 3 * 151          12       151        12    12
  1908     2^2 * 3^2 * 53         18       106        18    18
  10188    2^2 * 3^2 * 283        18       566        18    18
  10548    2^2 * 3^2 * 293        18       586        18    18
  11268    2^2 * 3^2 * 313        18       626        18    18
  12252    2^2 * 3 * 1021         12       1021       12    12
  12612    2^2 * 3 * 1051         12       1051       12    12
  12708    2^2 * 3^2 * 353        18       706        18    18
  13428    2^2 * 3^2 * 373        18       746        18    18
  14052    2^2 * 3 * 1171         12       1171       12    12
  14412    2^2 * 3 * 1201         12       1201       12    12
  15138    2 * 3^2 * 29^2         18       841        18    18
  18108    2^2 * 3^2 * 503        18       1006       18    18
  21984    2^5 * 3 * 229          24       916        24    24
  26480    2^4 * 5 * 331          20       1324       20    20
  27920    2^4 * 5 * 349          20       1396       20    20
  29360    2^4 * 5 * 367          20       1468       20    20
  39996    2^2 * 3^2 * 11 * 101   36       1111       36    36

A Special Class of Twin Primes

I was born in the year 1949 and am very aware of the fact that this number is prime and that it forms a twin prime pair with 1951. Today I turned 27751 days old and noticed that this number and 27749 also form a pair of twin primes. I asked myself the question: how many twin prime pairs are there in the range up to 40000 that end in the digits 49 and 51.

This is an easy question to answer. The result appears below (permalink):

• 149 and 151
• 1049 and 1051
• 1949 and 1951
• 2549 and 2551
• 4049 and 4051
• 4649 and 4651
• 5849 and 5851
• 6449 and 6451
• 7349 and 7351
• 7949 and 7951
• 11549 and 11551
• 14249 and 14251
• 14549 and 14551
• 16649 and 16651
• 20549 and 20551
• 26249 and 26251
• 27749 and 27751
• 28349 and 28351
• 33149 and 33151
• 33749 and 33751
• 34649 and 34651

There are 21 pairs and these are:

(149, 151), (1049, 1051), (1949, 1951), (2549, 2551), (4049, 4051), (4649, 4651), (5849, 5851), (6449, 6451), (7349, 7351), (7949, 7951), (11549, 11551), (14249, 14251), (14549, 14551), (16649, 16651), (20549, 20551), (26249, 26251), (27749, 27751), (28349, 28351), (33149, 33151), (33749, 33751), (34649, 34651)

Plotted, the point pairs appear as shown in Figure 1, forming a perfectly straight line:

Figure 1: permalink

It's interesting to note that even when the 49 and 51 pairs are not both prime, they always seem to be \( \textbf{relatively prime} \) just as the initial numbers, 49 and 51, are. My conjecture is that adding an equal number of additional digits to the left of 49 and 51 does not change this relative primeness. In other words:$$ gcd( \dots \text{xxx}49 \text, \dots \text{xxx}51)=1$$These 49 and 51 number pairs will always be interesting because they surround the midpoint of centuries as reckoned by the span from one 0 to the next. Thus the midpoint of the numbers from 0 to 100 is 50, the midpoint of the numbers from 100 to 200 is 150 etc.

The algorithm is easily adapted to search for other prime pair digit endings but 49 and 51 are the ones that attract my interest. If we extend our search further we can find some interesting special cases. For example, there are pairs of primes that both start and end in 49 and 51 (the pairs can longer be twin primes of course). Up to one million, there are only four pairs and these are (permalink):

• 49549 and 51551
• 491149 and 511151
• 494749 and 514751
• 499549 and 519551

Read about the unrelated and largely non-mathematical \( \textbf{49 - 51 principle} \) (link). Here is a summary of the report (link) that Gemini Deep Research prepared to the prompt: 

What is significant, mathematically and otherwise, about the 51 : 49 ratio. Can you create a report that highlights the most interesting information associated with this important ratio?

The 51:49 ratio, while mathematically representing a near-even split with decimal equivalents of 0.51 and 0.49 (or 51% and 49%), holds a significance that extends far beyond its basic numerical properties. Its proximity to perfect equality often creates an initial perception of balance, yet this subtle deviation carries substantial weight in numerous real-world contexts. In demographics, it appears as a natural tendency in human birth rates. In voting, it frequently marks the threshold of a narrow but often decisive majority. In business, it defines power dynamics in equity partnerships and serves as the foundation for a cultural principle promoting generosity. Even in seemingly random events like a coin toss, a slight 51:49 bias has been observed.

The power of this slight imbalance is evident in competitive scenarios where it often dictates victory and control. In business, it highlights the delicate interplay between majority rule and the rights of the minority. Psychologically, a 51:49 split is perceived as close and can influence the emotional responses to wins and losses, as well as the sociological dynamics of near-even divisions within society. While not uniquely tied to major historical events in its precise form, the concept of a narrow majority it represents has been historically significant. Moreover, the "51/49 Principle" has emerged as a contemporary cultural phenomenon. Comparisons with other near-even ratios like 50.5:49.5 and 52:48 further underscore the subtle but important nuances associated with small numerical differences around the midpoint. Existing research across various fields confirms that the 51:49 ratio is not just a theoretical concept but a subject of empirical study with real-world implications.

In conclusion, the 51:49 ratio, though seemingly representing a minimal imbalance, often acts as a critical threshold or a subtle but important bias with significant consequences across a diverse range of fields. Its significance lies not just in its mathematical representation but in its ability to define outcomes, shape relationships, and influence perceptions in the complex tapestry of the real world. 

A Seeming Coincidence

A friend of mine who turns 49 this year (2025) was born (of course) in 1976. I noticed that I will turn 76 this year and that I was born in 1949. If we ignore the '19' in our years of birth then our years of birth can be written as '76 and '49. The "coincidence" then is that we have this '76 and 49 versus '49 and 76 switching of the numbers. 

As I explained to my friend:

Taken to the extreme, I will turn 99 in 2048 in the unlikely event that I make it that far. Someone born in 1999 will turn 49 in this same year and so we have '99 and 49 versus '49 and 99. It doesn't hold in my case for people born in 2000 and beyond.