When Dates and Days Align

I put this little problem to Gemini:

A person is born on April 3rd 1949. The day that he is born is reckoned as Day 0 and for every subsequent day 1 is added and this count becomes his diurnal age. Let the number associated with any date be determined as DDMYY where DD is the number associated with the month(1 to a maximum of 31), M is the month (from 1 to 9) and YY is the last two digits of the current year. Thus a date of 14th September 2026 would become 14926. Is there any date, using this system, where the diurnal age equals the date?

Here was its response:

September 23, 2014, is the sole date where the diurnal age mathematically equates to its DDMYY numerical representation. 

That is interesting and so, on the 23rd of September 2014, I was 23914 days old. I got Gemini to write a program that would accept any birthdate as input using this prompt:

Can you write a program in SageMath that will accept any birthdate as input and from there calculate if there is a date where the diurnal age equals the date number using DDMYY format. Let’s realistically limit the person’s possible age to less than 100.  

Here is a permalink to the program and the result for the following birthdate:

• 23rd August 1964:
  Match found: 05 February 1979 | Diurnal Age / Date Number: 5279

However, I realised that I could be less restrictive with the date format and modified the prompt as follows:

I’d like to modify the program so the acceptable date format can be DMMYY, DDMYY or DMYY. This ensures that the resultant number derived from the date is always less than six digits. Everything else remains the same.

Here is a permalink to the modified program which produces the same results for the two dates examined above. Here are some additional dates:

• 23rd November 1980:
  Match found: 06 July 1999 | Diurnal Age: 6799 | Date Format: 6799
  Match found: 16 January 2025 | Diurnal Age: 16125 | Date Format: 16125
  
  
• 6th October 2002:
  Match found: 10 September 2032 | Diurnal Age: 10932 | Date Format: 10932

So just a little curiosity. Not all birthdates yield a result within the span of 100 years. You could ask the person born on the 6th October 2002: how many days old will you be on the 10th September 2032? Don't know? Here's a way to find out:

• 10th \( \rightarrow\) 10
• September \( \rightarrow\) 9
• 2032 \( \rightarrow\)  32
• Concatenation \( \rightarrow\) 10932 \( \rightarrow\) your diurnal age!