5 Result of te Riel was misstated, and better results exist

In reference to the Prime Number Theorem (then Conjecture) both Gauss and Riemann further conjectured that $\pi(n) < Li(n)$ (where $\pi(n)$ is the number of primes from $1$ to $n$ and $Li(n)$ is the logarithmic integral, $\int_2^n \frac{1}{ln(t)}dt$).

Although it has been proven that this does not hold (Littlewood), that there exists some $n$ such that $\pi(n) \geq Li(n)$, the first $n$ where this takes place is so huge no-one has worked it out yet (allegedly). The number is known as Skewes' Number. It happens somewhere is known to be between $6.62 \times 10^{370}$ 10^{14}$and$6.69 \times 10^{370}$1.39822\times 10^{316}$, and strongly believed to be about $1.397162914\times 10^{316}$. (Riele). References at the foregoing link.)

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In reference to the Prime Number Theorem (then Conjecture) both Gauss and Riemann further conjectured that $\pi(n) < Li(n)$ (where $\pi(n)$ is the number of primes from $1$ to $n$ and $Li(n)$ is the logarithmic integral, $\int_2^n \frac{1}{ln(t)}dt$).

Although it has been proven that this does not hold (Littlewood), that there exists some $n$ such that $\pi(n) \geq Li(n)$, the first $n$ where this takes place is so huge no-one has worked it out yet (allegedly). It happens somewhere between $6.62 \times 10^{370}$ and $6.69 \times 10^{370}$ (Riele).

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In reference to the Prime Number Theorem (then Conjecture) both Gauss and Riemann further conjectured that $\pi(n) < Li(n)$ (where $\pi(n)$ is the number of primes from $1$ to $n$ and $Li(n)$ is the logarithmic integral, $\int_2^n \frac{1}{ln(t)}dt$).

Although it has been proven that this does hold (Littlewood), that there exists some $n$ such that $\pi(n) \geq Li(n)$, the exact first $n$ where this takes place is so huge no-one has worked it out yet (allegedly). It is certainly larger than happens somewhere between $10^{16}$.6.62 \times 10^{370}$and$6.69 \times 10^{370}\$ (Riele).

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