The 11th anuvāka (paragraph) of Camakam, a collection of Vedic chant from the compendium of Kriśṇa Yajurveda’s Taittariya Samhita 4.7 is an interesting piece of text for it contains numbers explicitly.
The text of the 11th anuvāka of Camakam is:
Very briefly, the words in the above passage indicate first an odd number sequence starting from 1 going up till 33 whereafter the sequence resumes from 4, 8,12, 16, 20, 24, 28, 32, 36,40, 44, and 48.
Very recently I came across the video presentation by Sri K. Suresh (https://youtu.be/h6ERtEnpcvM) where he has explained the meaning and significance of the above sequence. Sri Suresh’s exposition can be summarized in the following table:
Here C1 is simply the sequence of natural numbers starting from 0. C2 is the square of the numbers in C1. C3 is obtained from entries of C2 in the following manner:
C3(i) = C2(i+1) — C2(i)
And C4 is obtained from entries in C3 in the following manner:
C4(i) = C3(i+1) + C3(i)
where ‘i’ is the position index.
As per the Vedic text given above, the “i” ranges from 1 to 17 in C3 and from 1 to 12 in C4.
Generally speaking, entries in C3 can be independently generated by the formula (2n + 1) [ Note: (n+1)² — n² = 2n + 1; n starts from 0] and entries in C4 can be generated by the formula 4(n+1) [Note: ((n+2)² — (n+1)² + ((n+1)² — n²)) = 4(n+1); n starts from 0]
Now, upon extending this to the next column C5 by adding C4 and C3, we get the following table:
Note that C5 can be independently generated by the formula 6n+5 [Note: ((n+1)² — n²) + 4(n+1) = 6n+5; n starts from 0]
It is known that there are 168 prime numbers between 1 and 1000. In the above table, C5 contains 166 entries of which only 85 are primes. Of the remaining 81, it is interesting to note that the factors are primes (multiples of 5 have a relatively larger number of factors) as shown in the last column.
In the following figure, I compare the actual number of primes and the primes emerging from the above method along with the non-primes (y-axis represent number). It is safe to conclude that the pattern of primes emerging from the above method closely follows the pattern of the actual number of primes (I have shown only primes less than 1000).
I am not sure whether this is of any significance, but it was fun doing this!
