Midpoint of the Year

On the 7th August 2021, I made a post title Key Points in the Year and Figure 1 is an extract from that post:

Figure 1

In that post I failed to mention that the exact midpoint of a common or non-leap year occurs at noon on the 2nd of July. As today is June 30th, that point is now not far off. In leap years, the midpoint occurs at midnight on the 2nd of July. This middle day can be written as \( \textbf{20260702} \) using YYYYMMDD format or \( \textbf{2026183} \) using the year and the number of days that have elapsed in it. 

Both formats uniquely define any date but I prefer the latter for the purposes of number analysis. Unfortunately, feeding both of these numbers into my daily number analysis program caused the wheels to spin and the analysis didn't complete. The program handles the five digit numbers associated with my diurnal age but seven and eight digit numbers are too much for my M1 Macbook Air's processor to handle.

However, 2026183 is prime and its reciprocal has a period of 2026182. It forms a twin prime pair with 2026181. It's interesting to consider how many times 183 leads to primes when concatenated with a range of years from 2000 to 2100. Here are the years:

2005, 2006, 2011, 2014, 2015, 2018, 2021, 2023, 2026, 2030, 2041, 2047, 2069, 2093, 2098

Of these, only 2005, 2026 and 2047 form twin prime pairs when both 181 and 183 are concatenated with them.