Problems & Puzzles: Conjectures
Conjecture 14.- Enoch Haga Observation about Palprimes
Patrick De Geest has been collecting the quantity of Palprimes of every (odd) length. Please see here.
The results of this collection up today are:
Enoch Haga had the inspired idea of multiplying this two columns to get a third one: the number of digits in the total palprimes of each length:
Can you see it? can you see the rate of growth of the numbers of the last column? its around 10 (ten) !!!
Enoch Haga wrote [Feb 28, 1999] to Patrick:
I'm surprised to see that the successive number of digits in the total of palprimes from 1-digit to 17-digits seems to increase by approximately a constant multiplier of 10. E.g. 45 is ~ 10*4 = 40 (45 actual), 465 is ~ 10*45 = 450 (465 actual), and so on [Sloane A039657]
Enoch, immediately realized the predictive capability of his discovery and added:
Therefore, it is easy to guess at the approximate number of 19-digit and 21-digit palprimes (I will not hazard a guess beyond that!). Simply divide the estimated total number of digits by the number of digits; thus 4597688420/19 = ~ 241.983.600 19-digit palprimes. The same procedure yields an estimate of ~ 2.189.370.676 21-digit palprimes.
Jud McCranie has obtained an explanation of the Enoch's observation. Here is his explanation:
P(D), the quantity of primes of D digits is given by:
D*Ppal(D) = (10 -D/(D-1)) * 10^((D+1)/2) /(9*ln(10))
This answer the two questions posed.