<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-infinitude-of-primes" target="_blank">Infinitude of Primes Exercise</a></p><p>There are infinitely many primes, though the number of primes within a fixed interval tend to decrease as the numbers get larger. For example, you can see how the number of primes decreases in each interval.</p>
<p>$$1-100: 25$$</p>
<p>$$101-200: 21$$</p>
<p>$$201-300: 16$$</p>
<p>So if the primes keep decresaing, how do we know they're infinite? You can show this through Proof by Contradiction. Imagine that the primes are finite. The only primes we have are $2$, $3$, and $5$. You then create an integer using these primes and adding $1$:</p>
<p>$$2 \times 3 \times 5 + 1 = 31$$</p>
<p>Notice that, by adding $1$, we created a new number that <strong><u>cannot</u></strong> share any of the prime factors of the number $1$ below ($30$). And in so doing we found a new prime: $31$.</p>
<p>So then we add $31$ to our list of "finite" primes: $2$, $3$, $5$, $31$. We then repeat the process:</p>
<p>$$2 \times 3 \times 5 \times 31 + 1 = 931$$</p>
<p>Now, $931$ is NOT prime, but its prime factors are completely different than the prime factors of the number below ($930$). With $931$, we have found TWO new primes not on our list: $7$ and $19$.</p>
<p>This process can be repeated forever. By adding $1$, you'll create a new number that always gives us more primes. Pretty frickin' cool! Euclid figured this shit out over 2,000 years ago. Dude was on another level. </p>