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| − | Find the largest prime below | + | Find the largest prime below $10^9$. How many primes numbers are there before this one? Which one is next? Compare to the prime-counting function $\pi(x)$ that counts how many primes there are below $x$ and which Gauss approximated to $x/\ln(x)$. |
<div class="mw-collapsible mw-collapsed" data-collapsetext="Solution" data-expandtext="Answer"> | <div class="mw-collapsible mw-collapsed" data-collapsetext="Solution" data-expandtext="Answer"> | ||
| − | Answer: This is 999,999,937 | + | Answer: This is $999\,999\,937$, the $50\,847\,534$th prime number, for which the prime-counting function $\pi(x)$ compares to Gauss' approximation $\pi(x)\approx x/\ln x=48\,254\,942$ to withing 5% accuracy. |
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| + | well, more things today? | ||
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<ul class="mw-collapsible mw-collapsed" data-collapsetext="I understand" data-expandtext="Click here for more information"> | <ul class="mw-collapsible mw-collapsed" data-collapsetext="I understand" data-expandtext="Click here for more information"> | ||
Find the largest prime below $10^9$. How many primes numbers are there before this one? Which one is next? Compare to the prime-counting function $\pi(x)$ that counts how many primes there are below $x$ and which Gauss approximated to $x/\ln(x)$.
Answer: This is $999\,999\,937$, the $50\,847\,534$th prime number, for which the prime-counting function $\pi(x)$ compares to Gauss' approximation $\pi(x)\approx x/\ln x=48\,254\,942$ to withing 5% accuracy.
well, more things today?