There is prime rate for borrowing money at the lowest change of currency in international commercialized banking. There is prime time for expecting radio and TV listeners and viewers with the highest number of people. Now there ought to be a prime tune as another way of looking at Euler-Gauss-Legendre prime number theorem by computational change of base, from natural to circular, from e to p.

At this point, there is no way of knowing for sure who started using the symbol p(x) denoting the number of primes less than the integer x. But then in 1841, Dirichlet (1805-59), who was a student of Gauss and Jacobi, started applying complex analysis for describing primes distribution. Earlier, Euler (1707-83), Gauss (1777-1855), Legendre (1752-1833), and others have already surmised that the limit of the product of p(x) with the ratio of the natural logarithm of x over x as x approaches infinity is unity and not zero, thus reasserting the mathematically proven conjecture that there are an infinite number of prime numbers first proved by Euclid in 300 B. C. and now called the fundamental theorem of arithmetic.

If x is the positive integer 3572 then the number of primes is exactly 500 and the largest happens to be the integer 3571. However, the natural logarithm of 3572 is approximately 8.18088942 and dividing 3572 by this number gives approximately 436.6277942 which is 63.37220584 short of the exact number 500. On the other hand, by simply a change of base, from e to p, the circular logarithm of 3572 is 7.146560113 and dividing 3572 by this new number gives 499.808850. This is within three hundredth (3/100) of a percent of error compared to one tenth (1/10) using the natural base. Incidentally, rounding to the nearest whole number, both e and p give the integer 3. Using base 3 the error is within 4/100 of a percent. There is no compelling reason for using the bases (e and p) other than the fact that they are irrationals that keep popping up frequently in all branches of analysis. Moreover, it seems paradoxical that analysis can be properly applied to number theory (a study of rationals) at all since integers reflect the quantum nature of rationals while analysis studies the continuum nature of irrationals.