guest@cp-base:~/home/number-theory$
templates/number-theory-basics.cpp
compilable
$cat templates/number-theory-basics

Number Theory Basics

Primality test, factorization, Euler's totient, divisors, integer sqrt, and perfect power check.

[Number Theory]|
Jul 9, 2026
|explanatory_notes.md|
#prime#factorization#totient#divisors#sqrt
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$cat source_code/
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#include <bits/stdc++.h>
using namespace std;
using ll = long long;

bool isPrime(ll n) {
    if (n < 2) return false;
    if (n < 4) return true;
    if (n % 2 == 0 || n % 3 == 0) return false;
    for (ll i = 5; i * i <= n; i += 6)
        if (n % i == 0 || n % (i + 2) == 0) return false;
    return true;
}

vector<ll> primeFactorization(ll n) {
    vector<ll> factors;
    while (n % 2 == 0) { factors.push_back(2); n /= 2; }
    for (ll i = 3; i * i <= n; i += 2)
        while (n % i == 0) { factors.push_back(i); n /= i; }
    if (n > 1) factors.push_back(n);
    return factors;
}

ll totient(ll n) {
    ll result = n;
    for (ll i = 2; i * i <= n; i++) {
        if (n % i == 0) {
            while (n % i == 0) n /= i;
            result -= result / i;
        }
    }
    if (n > 1) result -= result / n;
    return result;
}

vector<ll> divisors(ll n) {
    vector<ll> divs;
    for (ll i = 1; i * i <= n; i++) {
        if (n % i == 0) {
            divs.push_back(i);
            if (i != n / i) divs.push_back(n / i);
        }
    }
    return divs;
}

int countDivisors(ll n) {
    int cnt = 0;
    for (ll i = 1; i * i <= n; i++)
        if (n % i == 0) { cnt += 2; if (i == n / i) cnt--; }
    return cnt;
}

ll sumDivisors(ll n) {
    ll sum = 0;
    for (ll i = 1; i * i <= n; i++)
        if (n % i == 0) { sum += i; if (i != n / i) sum += n / i; }
    return sum;
}

ll isqrt(ll n) {
    ll s = (ll)sqrt((double)n);
    while (s > 0 && s * s > n) s--;
    while ((s + 1) * (s + 1) <= n) s++;
    return s;
}

bool isPerfectSquare(ll n) {
    if (n < 0) return false;
    ll s = isqrt(n);
    return s * s == n;
}

bool isPerfectPower(ll n, int base) {
    if (n <= 0) return false;
    if (n == 1) return true;
    ll p = 1;
    while (p < n) {
        p *= base;
        if (p == n) return true;
        if (p > n / base) break;
    }
    return false;
}
83 linesutf-8
$cat explanation_notes.md
notes_viewer --renderedmarkdown (math enabled)

Number Theory Basics

Functions

  • isPrime(n) — trial division with 6k±16k \pm 1 optimization, O(n)O(\sqrt{n})

  • primeFactorization(n) — returns all prime factors with repetition

  • totient(n) — Euler's φ(n)=n∏p∣n(1−1/p)\varphi(n) = n \prod_{p \mid n} (1 - 1/p)

  • divisors(n) — all divisors (unsorted)

  • countDivisors(n) / sumDivisors(n) — count or sum of divisors

  • isqrt(n) — integer square root, adjusted from floating-point to avoid precision bugs for large nn

  • isPerfectSquare(n) — uses isqrt, safe for full ll range

  • isPerfectPower(n, base) — integer-only loop, no floating-point

    • Bug Fixes vs Original

  • isqrt adjusts ±1\pm 1 from (ll)sqrt(n) — the original silently fails for large perfect squares like 101810^{18}

  • isPerfectPower uses multiplication loop instead of log to avoid precision errors

  • primeFactorization uses ll loop variable (original used int, overflows for large nn)

    • Complexity

  • All functions — O(n)O(\sqrt{n}) except isPerfectPower which is O(log⁡basen)O(\log_{\text{base}} n)