Longest repeated sequence in the prime counting function

**Loren Booda** · Jun19-09, 02:14 AM

Is there a longest repeated sequence (congruency) in the prime counting function $LaTeX Code: \\pi (x)$ (that which gives the number of primes less than or equal to x)?

Recall that $LaTeX Code: \\pi (x)$ , although infinite, may not be random, and itself starts out with an unrepeated sequence $LaTeX Code: \\pi (2)=1$ and $LaTeX Code: \\pi (3)=2$ (with a "slope" of 1).

**CRGreathouse** · Jun19-09, 03:16 AM

I really can't tell what you mean. Could you explain what you mean more carefully, maybe with an example?

**Simon-M** · Jun19-09, 04:54 AM

There are arbitrarily large gaps in the prime numbers. This means that $LaTeX Code: \\pi(n)$ can be constant over an arbitrarily large interval.

Consider the n-1 numbers

where $LaTeX Code: k=2,3,\\cdots n$

**Loren Booda** · Jun19-09, 11:23 AM

I think that Simon-M answered my question, and with a basic example - that the prime counting function as graphed can repeat itself indefinitely (such as when constant over an arbitrarily large interval). Another example would include the interjection of one prime into such an arbitrarily large sequence, which then could be repeated.