Day 3 — Number Theory
Time: ~45 min · Topic: number theory (primes, modular arithmetic, divisibility)
The problem
Find all prime numbers p such that p^2 + 2 is also prime.
Hint: What happens when you divide p^2 + 2 by 3 for different primes? Think about what every prime greater than 3 looks like mod 3.
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Test small primes. For p = 2: 2^2 + 2 = 6 = 2 x 3, not prime. For p = 3: 3^2 + 2 = 11, which IS prime. So p = 3 works.
- Now consider any prime p > 3. Every such prime is NOT divisible by 3, so p must satisfy either p ≡ 1 (mod 3) or p ≡ 2 (mod 3).
- Check both cases. If p ≡ 1 (mod 3), then p^2 ≡ 1 (mod 3), so p^2 + 2 ≡ 3 ≡ 0 (mod 3). If p ≡ 2 (mod 3), then p^2 ≡ 4 ≡ 1 (mod 3), so p^2 + 2 ≡ 3 ≡ 0 (mod 3). Either way, 3 divides p^2 + 2.
- Since p > 3 implies p^2 + 2 > 11 > 3, and we just showed 3 divides p^2 + 2, it must be composite. Therefore p = 3 is the ONLY prime where p^2 + 2 is also prime.
Answer: p = 3 is the only prime such that p^2 + 2 is also prime (giving 11).
Why this matters
This proof technique — checking residues mod 3 — is the same idea behind how the AKS algorithm proved that primality testing can be done in polynomial time, earning its creators the 2006 Gödel Prize.
This day in math: The Mathematician Who Broke Infinity Into Pieces
April 23, 1858 · Max Planck
On April 23, 1858, Max Planck was born in Kiel, Germany. In 1900, he stunned the scientific world by proving that energy isn't continuous — it comes in tiny, discrete packets called "quanta." His deceptively simple equation E = hf, where energy equals Planck's constant times frequency, shattered centuries of assumption that nature behaves smoothly.
Why it matters: Planck's insight — that the universe is fundamentally discrete, not continuous — launched quantum mechanics and reshaped both physics and mathematics. Every piece of modern technology, from smartphones to MRI machines, traces back to this one equation.
Did you know?
There are more numbers between 0 and 1 than there are counting numbers (1, 2, 3, 4, ...) stretching to infinity. Not all infinities are the same size — some infinities are literally BIGGER than others!
In 1891, Georg Cantor proved this with his famous "diagonal argument." He showed that no matter how you try to pair up every decimal between 0 and 1 with a counting number, you can always build a new decimal that isn't on your list. This means the infinity of real numbers is "uncountable" — a fundamentally larger type of infinity than the "countable" infinity of 1, 2, 3, 4, and so on.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact