The March of Time

I've written about what I term AD and BC numbers in a post titled, quite sensibly, AD and BC Numbers. I was reminded of them because my diurnal age today, 27068, converts to 69BC in hexadecimal. For some time now, these BC numbers have been occurring every 256 days.

24764 --> 60bc

25020 --> 61bc

25276 --> 62bc

25532 --> 63bc

25788 --> 64bc

26044 --> 65bc

26300 --> 66bc

26556 --> 67bc

26812 --> 68bc

27068 --> 69bc

However, this regular march of time is now at an end because if 256 is added to 27068, the resultant number (27324) is 6ABC. The decimal equivalent of 70AD is 28845, representing a jump of 1792 or 7 x 256 days.

Similarly, I recently turned 27053 days old which is 69AD in hexadecimal. Notice the difference of 14 days between 69AD and 69BC. This number was also the end of a run of numbers differing by 256 days.

 

24749 --> 60ad

25005 --> 61ad

25261 --> 62ad

25517 --> 63ad

25773 --> 64ad

26029 --> 65ad

26285 --> 66ad

26541 --> 67ad

26797 --> 68ad

27053 --> 69ad

The decimal equivalent of 70AD is 28845, again a jump of 1792 or 7 x 256 days. Taken over a long enough time period, this represents an average advance of a little over 395.6 days, about a month longer than the solar year. Both AD and BC dates are advancing at this average rate.

However, it will almost five years before I encounter another hexadecimal AD and BC number so I thought it important to mark the fact in this post. While exercises like this may seem frivolous, they nonetheless provide an opportunity to work with hexadecimal numbers and convert from decimal to hexadecimal and vice versa. I'm always thinking in terms of the former mathematics teacher who I once was.

How can hexadecimal numbers be made interesting for students? These AD and BC numbers are a way of doing this. The question could be asked of students:

Find out your diurnal age and determine when you will next have a connection to AD or BC year (via decimal to hexadecimal conversion). What was significant about that year.

See my post titled 69BC for details on what was significant about this year in history. Students could be shown how to determine their diurnal age using Wolfram Alpha and they could also use it to convert between decimal and hexadecimal. Overall, a useful and interesting exercise.

AD and BC Numbers

In my quest to find interesting properties for the successive numbers that constitute my diurnal age, I usually make use of the OEIS and Numbers Aplenty. Occasionally these resources throw up little of interest and, after exhausting additional resources, I'm left to come up with some creative angle of my own.

Today's number, 26541, presented such a challenge and I'm proud to say I rose to the occasion. While professional mathematicians will sneer, recreational mathematical enthusiasts may appreciate my resourcefulness. Here is what I came up with. Decimal numbers when converted to hexadecimal can end in AD and the thought struck me that such numbers would form a sequence. I've entered this sequence into my own private database of sequences. Here it is:

S037: Anno Domini (AD) numbers: decimal numbers that when converted to hexadecimal contain at least three digits satisfying the following criteria: 

• the second last digit is A
• the last digit is D
• all remaining digits are between 0 and 9

Up to 40000, the list of such numbers is as follows:

429, 685, 941, 1197, 1453, 1709, 1965, 2221, 2477, 4269, 4525, 4781, 5037, 5293, 5549, 5805, 6061, 6317, 6573, 8365, 8621, 8877, 9133, 9389, 9645, 9901, 10157, 10413, 10669, 12461, 12717, 12973, 13229, 13485, 13741, 13997, 14253, 14509, 14765, 16557, 16813, 17069, 17325, 17581, 17837, 18093, 18349, 18605, 18861, 20653, 20909, 21165, 21421, 21677, 21933, 22189, 22445, 22701, 22957, 24749, 25005, 25261, 25517, 25773, 26029, 26285, 26541, 26797, 27053, 28845, 29101, 29357, 29613, 29869, 30125, 30381, 30637, 30893, 31149, 32941, 33197, 33453, 33709, 33965, 34221, 34477, 34733, 34989, 35245, 37037, 37293, 37549, 37805, 38061, 38317, 38573, 38829, 39085, 39341

All the numbers differ by 256 except for a regular jump of 1792 or 7 x 256 after every 9th number. In fact, all the numbers equal 173 mod 256 and 173 is the decimal value of AD. The first number in the sequence, 429 = 256 + 173, converts to 1AD while the last number, 39341 = 9 x 4096 + 6 x 256 + 173, converts to 99AD. The number associated with my diurnal age, 26541, converts to 67AD. Here is a permalink to the code that I used in SageMathCell.

The logical extension of course is to identify BC numbers and that is what I've done in another private sequence:

S038: Before Christ (BC) numbers: decimal numbers that when converted to hexadecimal contain at least three digits satisfying the following criteria: 

• the second last digit is B
• the last digit is C
• all remaining digits are between 0 and 9

Up to 40000, the members of this sequence are:

444, 700, 956, 1212, 1468, 1724, 1980, 2236, 2492, 4284, 4540, 4796, 5052, 5308, 5564, 5820, 6076, 6332, 6588, 8380, 8636, 8892, 9148, 9404, 9660, 9916, 10172, 10428, 10684, 12476, 12732, 12988, 13244, 13500, 13756, 14012, 14268, 14524, 14780, 16572, 16828, 17084, 17340, 17596, 17852, 18108, 18364, 18620, 18876, 20668, 20924, 21180, 21436, 21692, 21948, 22204, 22460, 22716, 22972, 24764, 25020, 25276, 25532, 25788, 26044, 26300, 26556, 26812, 27068, 28860, 29116, 29372, 29628, 29884, 30140, 30396, 30652, 30908, 31164, 32956, 33212, 33468, 33724, 33980, 34236, 34492, 34748, 35004, 35260, 37052, 37308, 37564, 37820, 38076, 38332, 38588, 38844, 39100, 39356

Again, all the numbers differ by 256 except for a regular jump of 1792 or 7 x 256 after every 9th number. Here, all the numbers equal 188 mod 256 and 188 is the decimal value of BC. The first number in the sequence, 444 = 256 + 188, converts to 1BC while the last number, 39356 = 9 x 4096 + 9 x 256 + 188, converts to 99BC. Here is a permalink to the code that I used in SageMathCell. 

I know that BCE and CE can be used instead of BC and AD but I refuse to use this system. Anyone so inclined however, could modify my existing algorithms or create algorithms of their own in order to generate hexadecimal numbers ending in BCE and CE.

The numbers don't have to be necessarily hexadecimal because the letters used in my system are A, B, C and D and thus will work for bases 14 and 15 and 17 up to 36. I just chose hexadecimal because of its widespread use in technology. 

It's interesting that 67 AD has associations with Christianity because this year marked the end of St. Paul's journeying around the Mediterranean. See Figure 1.

Figure 1: source

This post marks my 100th post for the year 2021, an annual record that I've never even approached before. Previously, my annual totals were:

• 2020: 68 posts
• 2019: 56 posts
• 2018: 68 posts
• 2017: 36 posts
• 2016: 42 posts
• 2015: 12 posts

I started this blog in September of 2015, shortly after my retirement from teaching.