Is this right?
The Prime Counting Function (PCF) says the number of primes less than or equal to N is given by N/ln(N), where ln(N) is the natural logarithm of N. For example the Prime Counting Function suggests 21.715 primes less than or equal to 100; suggesting a probability of any single integer between 1 and 100 being prime is 21.715 / 100 = 0.21715.
For TLW’s ten-digit magic square the smallest and largest numbers are:
N = 1480028213, PCF = 70092602.11
N = 1480028129, PCF = 70092598.32
A range of 84.
It seems risky to subtract two large numbers and put any faith in the small remainder, but ignoring that, the PCF suggest just 3.79 prime numbers lie between 1480028129 and 1480028213. Giving a probability of any particular single number being prime as 3.79 / 84 = 0.04512. The probability of all, of any nine numbers from within that range, being prime is (3.79/84)^9 = 7.75 x 10^-13. That seems a vanishingly small probability and suggests a remarkable result to find the case that beats the odds.