Editing Number Theory (section)

== <span style="color: #FFFFFF;">Analyzing</span> ==
{| class="wikitable"
|+ Important Number Theory Results
! Result !! Statement !! Significance
|-
| Infinitely many primes || No largest prime || Proved by Euclid c.300 BCE; multiple proofs
|-
| Fermat's Little Theorem || a^p ≡ a (mod p) || Foundation of RSA, primality tests
|-
| Quadratic Reciprocity || Law governing ±1 residuosity || Gauss's "gem"; proved ~8 ways
|-
| Prime Number Theorem || π(n) ~ n/ln(n) || Proved 1896; Riemann Hypothesis refines it
|-
| Fermat's Last Theorem || No x^n+y^n=z^n (n>2) || 358 years to prove; used elliptic curves
|}

'''Famous open problems''': Riemann Hypothesis (1859): all non-trivial zeros of ζ(s) have Re(s) = 1/2 — implies the most precise distribution of primes. Goldbach's Conjecture: every even number > 2 is the sum of two primes — verified to 4×10¹⁸, unproven. Twin Prime Conjecture: infinitely many pairs of primes differing by 2 — Zhang (2013) proved bounded gaps. Collatz Conjecture: starting from any positive integer, repeatedly dividing even numbers by 2 and multiplying odd numbers by 3+1 always reaches 1.
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