###### tags: `Number Theory` `NT01`
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L03.5 Prime Distribution and Weekness of Symmetrical Key
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Last Week
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- Primes, FTA, Factorisation
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This week
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- Prime Distritution
- Idea of mod + permutation
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1. Magic Primes
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1.1 How many are there?
<font size = 5>FTA tells us that Primes are **cornerstones**. </font>
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So how many such cornerstones there in the Natual Number's Regime?
The answer is infinitely many!
**Idea of proof (from Euclid again):**
if we have finite primes, says, $p_1$ and $p_2$, then, multiply them together, we get $p_1 p_2$, which is a composite number. But how about $1+p_1 p_2$? Is it a prime or composite?
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1.2 Number of Primes given a range
<font size = 5>Even though we do not know exactly how many primes there, we know approximately, number of primes given a range. </font>
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$$\pi(n) \approx \frac{n}{ln(n)}$$
Where n is a large number refers to the range, e.g. within 100000, etc.
$\pi(n)$ refers to the approx number of primes given n as upperbound.
$ln(n)$ refers to the natural logarithms of n.
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