Birthday Digits

Gemini's Infographic Summary of the Content in this Post

I recently turned 28077 days old and with my 77th birthday rapidly approaching I wondered if it's possible for a person's birthday to fall in such a way that their age in years matched the last two digits of their diurnal age. I prompted Gemini:

Suppose someone is born on April 3rd 1949. Let's count this as day 0. April 4th 1949 then counts as day 1 and so on. This progressive number we term the person's diurnal age as opposed to their age in years. As the person ages (let's say from 1 to 100) is there a birthday such that the last two digits of their diurnal age equals their age in years. Can you write a program in SageMath to determine this. The program should run in SageMathCell or a Jupyter notework.

The program (permalink) came up with only one age, 14, and an associated diurnal age of 5114 for person's born within a certain date range. For example for persons born in 2012 who are turning 14 this year (2026), only those with birthdates between the 1st January and 28th February will satisfy. Let's check for somebody turning 14 today (16th February 2026) using Wolfram Alpha. See Figure 1.

Figure 1

As can be seen, someone turning 14 years old today will be 5114 days old, satisfying the requirement. I queried Gemini as to whether there might be ages other than 14 that satisfy and this was the response:

Yes, I am sure. Age 14 is the only age in a normal human lifespan (1–100+) where this mathematical coincidence occurs. We can prove this by checking the mathematical constraints for every single age.

The Logic: Why it's only 14

For a match to happen at Age $N$, the following equation must be true: $$(64 \times N) + \text{Leap Days} \quad \text{must be divisible by } 100$$

At Age 14: $64 \times 14 = 896$. The remainder is 96. To reach the next 100, we need 4 more. Can a 14-year-old have lived through 4 leap days? Yes (e.g., born Jan 1948). Result: Match.

At Age 39 (The next closest candidate): $64 \times 39 = 2496$. The remainder is 96. To reach the next 100, we again need 4 more. Can a 39-year-old have lived through only 4 leap days? No. In 39 years, you must experience 9 or 10 leap years. Result: Impossible.

Gemini also explained it earlier in a slightly different way so I'll include that here:

The Math Behind the Age 14 Match

For a match to occur, we need: $$\text{Total Days} \pmod{100} = \text{Age}$$

Let's look at Age 14:

Standard Days: $14 \text{ years} \times 365 \text{ days/year} = 5110 \text{ days}$.

The Remainder: $5110 \pmod{100} = 10$.

The Gap: The remainder is 10, but we need it to equal the age, 14.

The Solution: We need exactly 4 extra days (leap days) to bridge the gap from 10 to 14 ($5110 + 4 = 5114$).

Therefore, any person who experiences exactly 4 leap years by their 14th birthday will have a diurnal age of 5114. Since 5114 ends in 14, this is a perfect match.

Why April 3, 1949 Misses the Match

Let's apply this to your specific date: April 3, 1949.

Period: April 3, 1949 to April 3, 1963.

Leap Years Encountered: 1952, 1956, 1960.

Total Leap Days: 3.

Total Diurnal Age: $5110 + 3 = 5113$.

Result: $5113$ ends in 13. The age is 14. (Missed by 1 day).

Who Gets the Match?

To get 4 leap days in 14 years, you must be born just before a leap day, so that you catch the first one immediately, and then three more over the next 12 years.

This generally happens to people born in January or February of a Leap Year (or late in the year prior to a leap year).

28003: A Lesson Learned

One of the limitations of the free version of the Airtable online database is that it only allows about 1000 records per database. Once the limit is reached, you need to create a new database to accommodate your needs. I've been using Airtable for years now and I am into my third database:

• Diurnal Age Part 1
• 1198 records (over the limit)
• started on March 17th 2017
• closed on June 26th 2021
  
  
• Diurnal Age Part 2
• 923 records
• started on June 27th 2021
• closed on January 2nd 2024
  
  
• Diurnal Age Part 3
• 704 records currently
• started on January 3rd 2024
  
  

This is an impressive combined database containing about 3000 records. However, the records are in three disconnected databases and once I move on to a new database, I've tended to ignore the earlier databases. However, I realised the folly of doing this when confronted with something interesting to say about the number associated with my diurnal age today: \( \textbf{28003}\).

It's not dark yet but it's getting there

The OEIS, Mathematical Meanderings (my Mathematics blog), Bespoken for Sequences and Diurnal Age Part 3 (in short all my usual sources) had nothing useful to contribute. It then occurred to me to look back at my Diurnal Age Part 2 database and I was pleasantly surprised at what I found therein. Let's begin:

• \( \textbf{28003} \) is a member of OEIS A179248: numbers that have 8 terms in their Zeckendorf representation: [3, 8, 89, 233, 610, 2584, 6765, 17711]. There is in face a cluster of sequence members nearby, namely 27800, 27802, 27804, 27807, 27808, 27809 and 27811.
  
  
• \( \textbf{28003}\) is a member of OEIS A157344: semiprimes that are the product of two distinct Sophie Germain primes. Approximately 2.52% of numbers in the range up to 40,000 satisfy this criterion. Here the primes are 41 and 683 and we have:
• 41 x 2 + 1 = 83 which is prime
• 683 x 2 + 1 = 1367 which is prime
  
  
• \( \textbf{28003}\) is a member of OEIS A082957: numbers \(n\) such that \( \sigma(2n) < \sigma(2n+1) \). These numbers total 5.52% in the range up to 40,000. There are 2301 primes and 3169 composites among the 5470 first terms. Here we have:
• \( \sigma(2 \times 28003) = \sigma(56006) = 86184\)
• \( \sigma(2 \times 28003 + 1) = \sigma(56007) = 94848\)
  
  
• \( \textbf{28003} \) is a member of OEIS A134252: positions of 2 after decimal point in decimal expansion of \( \frac{1}{\pi}\). 

Diurnal Age Part 1 yielded no results but as can be seen Diurnal Age Part 2 revealed membership of \( \textbf{28003}\) in four different OEIS sequences.

Gray Code

I'd never heard of this before. I came across it when looking at the properties associated with my diurnal age of 27104. One of its properties is that it's a member of OEIS A265385:

A265385

Sequence defined by a(1) = a(2) = 1 and a(\(n\)) = gray(a(\(n\)-1) + a(\(n\)-2)), with gray(\(m\)) = A003188(\(m\)).

Here's what Wikipedia has to say on the topic:

The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).

For example, the representation of the decimal value "1" in binary would normally be "001" and "2" would be "010". In Gray code, these values are represented as "001" and "011". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.

Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems.

Table 1 shows the relationships between binary, gray code and decimal.

Table 1: source

To understand how this OEIS arises, I needed a way to convert binary to gray code and this site provided some Python code to achieve this. I modified it slightly to accept decimal input (permalink). Armed with this, I began to investigate the sequence. 

The initial values are 1 and 1 and these sum to 2. The binary code for 2, as can be seen from Table 1, is 0010 which the Gray code converts to 0011 and this has a decimal equivalent of 3. Proceeding like this we have:

• 1 first seed value
• 1 second seed value
• 1 + 1 = 2 --> 3
• 1 + 3 = 4 --> 6
• 3 + 6 = 9 --> 13
• 6 + 13 =19 --> 26
• 13 + 26 = 39 --> 52
• 26 + 52 = 78 --> 105
• 52 + 105 = 157 --> 211
• 105 + 211 = 316 --> 418
• 211 + 418 = 629 --> 847
• 418 + 847 = 1265 --> 1673
• 847 + 1673 = 2520 --> 3380
• 1673 + 3380 = 5053 --> 6755
• 3380 + 6755 = 10135 --> 13404
• 6755 + 13404 = 20159 --> 27104

The OEIS comments note that:

This recurrence is reminiscent of Fibonacci's, except that the result of each step is passed through the binary-reflected Gray code mapping, which introduces a degree of pseudo-randomness. 

However, the ratio of successive terms doesn't approach the Golden Ratio but instead fluctuates around 2 with a variation of about 1% above and below. Here are the successive ratios of \( \frac{(n+1)^{th} \text{ term}}{n^{th} \text{ term}}\):

3.0000000, 2.0000000, 2.1666667, 2.0000000, 2.0000000, 2.0192308, 2.0095238, 1.9810427, 2.0263158, 1.9752066, 2.0203228, 1.9985207, 1.9843079, 2.0220830, 1.9752804, 2.0203033, 1.9986779, 1.9841292, 2.0220523, 1.9752920, 2.0202956, 1.9986750, 1.9841298, 2.0220479, 1.9752930, 2.0202950, 1.9986749, 1.9841299, 2.0220478, 1.9752930, 2.0202950, 1.9986725

There's lots of YouTube videos explaining about Gray code and I'm happy to have finally stumbled upon this clever manipulation of binary code. The OEIS A003188 referred to above lists the decimal equivalents of the Gray code for integers \(n\). The result is a permutation of the order of the cardinal numbers as shown for \(n\) up to 70:

• 0 --> 0
• 1 --> 1
• 2 --> 3
• 3 --> 2
• 4 --> 6
• 5 --> 7
• 6 --> 5
• 7 --> 4
• 8 --> 12
• 9 --> 13
• 10 --> 15
• 11 --> 14
• 12 --> 10
• 13 --> 11
• 14 --> 9
• 15 --> 8
• 16 --> 24
• 17 --> 25
• 18 --> 27
• 19 --> 26
• 20 --> 30
• 21 --> 31
• 22 --> 29
• 23 --> 28
• 24 --> 20
• 25 --> 21
• 26 --> 23
• 27 --> 22
• 28 --> 18
• 29 --> 19
• 30 --> 17
• 31 --> 16
• 32 --> 48
• 33 --> 49
• 34 --> 51
• 35 --> 50
• 36 --> 54
• 37 --> 55
• 38 --> 53
• 39 --> 52
• 40 --> 60
• 41 --> 61
• 42 --> 63
• 43 --> 62
• 44 --> 58
• 45 --> 59
• 46 --> 57
• 47 --> 56
• 48 --> 40
• 49 --> 41
• 50 --> 43
• 51 --> 42
• 52 --> 46
• 53 --> 47
• 54 --> 45
• 55 --> 44
• 56 --> 36
• 57 --> 37
• 58 --> 39
• 59 --> 38
• 60 --> 34
• 61 --> 35
• 62 --> 33
• 63 --> 32
• 64 --> 96
• 65 --> 97
• 66 --> 99
• 67 --> 98
• 68 --> 102
• 69 --> 103
• 70 --> 101

Honaker: Primes and Problem

Figure 1

G. L. Honaker Jr is a rather shadowy figure who, as we learn from his LinkedIn profile, is a Maths/Science graduate who has been teaching for 28 years. He lives in Bristol, Virginia, and must be around 50 years old. There's not much other biographical data about him, except that he is the co-author of a book that I'll talk about later.

The reason I mention him is that his name came up today as part of my diurnal age number investigation. Today I'm 26119 days old and this turns out to be a Honaker prime. It's pretty cool to have a type of prime number named after you. So what is a Honaker prime? It's defined as a prime \(p_n\) whose index \(n\) and \(p_n\) itself have the same sum of digits. For example, \(p_{32}=131\) is a Honaker prime because \(3+2=1+3+1\).

Unlike primeness, the property that defines a Honaker prime is base specific. For example in base 2, 32 becomes 100000 and 131 becomes 10000011, but the sums of their digits are different. The smallest prime which is Honaker in all the bases from 2 to 10 is  \(p_{277308991}= 5949670231\). I'm thankful to NumbersAplenty for this information.

Not only does Honaker have a type of prime named after him, he also has an eponymous problem. Wolfram MathWorld explains that Honaker's problem asks for all consecutive prime number triples \((p,q,r)\) with \(p<q<r\) such that \(p|(qr+1)\). Caldwell and Cheng (2005) showed that the only Honaker triplets for \(p<=2×10^{17} \text{ are }(2, 3, 5), (3, 5, 7) \text{ and } (61, 67, 71)\). It is conjectured that these three triplets may be the only such triplets with this property.

Interestingly, I'm currently 71 years old and 71 forms part of a Honaker triplet in addition to the fact that my diurnal age (26119) is a Honaker prime. Now getting back to Honaker's book that is available on Amazon. There is an accompanying website that boasts that "there are currently 25464 curios corresponding to 18548 different numbers in our database, that leaves an infinite number for you to discover!"

Naturally I typed in 26119 and discovered that "The number of Honaker primes less than or equal to 26119 is the smallest Honaker prime." This indeed true as the smallest Honaker prime is 131 and there are exactly 131 Honaker primes less than or equal to 26119. I was impressed. Clearly, this is a book to buy and a website to visit for anyone seriously interested in prime numbers.