https://teddyrookbook.com/math/number-theory/Recent content in Number Theory on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/number-theory/divisibility/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/divisibility/a ∣ b means b = ak for some integer k; if a divides b and c, it divides every combination mb + nc. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/division-algorithm/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/division-algorithm/For a positive divisor b, every integer a can be written as a = bq + r with 0 ≤ r < b, even when a is negative. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/number-bases/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/number-bases/Convert to base b by dividing by b again and again, then read the remainders from last to first. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/euclidean-algorithm/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/euclidean-algorithm/gcd(a, b) = gcd(b, r), where r is the remainder when a is divided by b; repeat until the remainder is 0. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/bezout/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/bezout/Work the Euclidean algorithm backward to write gcd(a, b) = ax + by. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/linear-diophantine/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/linear-diophantine/ax + by = c has integer solutions exactly when d = gcd(a, b) divides c, and then x = x₀ + (b/d)t, y = y₀ − (a/d)t. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/primality-sieve/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/primality-sieve/n is prime when no prime up to √n divides it; the sieve crosses out multiples to list every prime up to N. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/unique-factorization/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/unique-factorization/Every integer above 1 is a product of primes in exactly one way, because a prime that divides ab divides a or b (Euclid's lemma). Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/gcd-lcm-factorizations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/gcd-lcm-factorizations/Take the smaller exponent of each prime for the gcd and the larger for the lcm; then gcd(a, b) · lcm(a, b) = ab. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/divisor-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/divisor-functions/If n = p₁^a₁ ⋯ pₖ^aₖ, then n has (a₁ + 1)⋯(aₖ + 1) divisors, and their sum σ(n) is a product of geometric sums. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/perfect-numbers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/perfect-numbers/n is perfect when σ(n) = 2n; whenever 2ᵖ − 1 is prime, 2ᵖ⁻¹(2ᵖ − 1) is perfect. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/factorial-primes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/factorial-primes/The exponent of p in n! is ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + ⋯, so 100! ends in 24 zeros. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/infinitely-many-primes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/infinitely-many-primes/Euclid's argument: p₁p₂⋯pₖ + 1 has a prime factor that is not on the list p₁, …, pₖ. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/prime-counting/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/prime-counting/The prime number theorem: there are about x/ln x primes up to x. Primes thin out, and n! + 2, …, n! + n is a run of n − 1 composite numbers. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/congruence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/congruence/a ≡ b (mod n) when n ∣ (a − b), that is, when a and b leave the same remainder on division by n. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/modular-arithmetic/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/modular-arithmetic/Reduce first, then add, subtract and multiply: the result is the same as reducing at the end. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/divisibility-tests/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/divisibility-tests/Since 10 ≡ 1 (mod 9) and 10 ≡ −1 (mod 11), digit sums and alternating digit sums test divisibility by 9 and by 11. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/modular-powers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/modular-powers/Square again and again, multiply the powers that match the binary digits of the exponent, and reduce after every step. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/modular-inverses/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/modular-inverses/a has an inverse mod n exactly when gcd(a, n) = 1; find it with the extended Euclidean algorithm. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/linear-congruences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/linear-congruences/ax ≡ b (mod n) has no solution unless d = gcd(a, n) divides b, and then it has exactly d solutions mod n. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/chinese-remainder/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/chinese-remainder/When m and n are coprime, the pair x ≡ a (mod m), x ≡ b (mod n) has exactly one solution mod mn. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/fermat-little/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/fermat-little/If p is prime and p ∤ a, then aᵖ⁻¹ ≡ 1 (mod p), so exponents can be reduced mod p − 1. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/wilson/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/wilson/An integer p > 1 is prime exactly when (p − 1)! ≡ −1 (mod p): the factors from 2 to p − 2 pair up with their inverses, leaving 1 · (p − 1) ≡ −1. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/euler-phi/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/euler-phi/φ(n) counts the k from 1 to n with gcd(k, n) = 1; φ(pᵏ) = pᵏ − pᵏ⁻¹, and φ(mn) = φ(m)φ(n) when gcd(m, n) = 1. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/euler-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/euler-theorem/If gcd(a, n) = 1, then a^φ(n) ≡ 1 (mod n); use it to find the last digits of huge powers. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/pseudoprimes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/pseudoprimes/If aⁿ⁻¹ ≢ 1 (mod n) for some a with 1 < a < n, then n is composite; passing the test does not prove that n is prime. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/order-mod-n/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/order-mod-n/The order of a mod n is the smallest k > 0 with aᵏ ≡ 1; it divides φ(n), and the powers of a repeat with that period. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/primitive-roots/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/primitive-roots/g is a primitive root mod p when its order is p − 1, so its powers reach every nonzero residue; check that g^((p − 1)/q) ≢ 1 for each prime q dividing p − 1. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/discrete-logs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/discrete-logs/With a primitive root g mod p, every nonzero residue is gᵏ for exactly one k mod p − 1; checking a k is easy, finding one is slow. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/affine-ciphers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/affine-ciphers/Encrypt letters with x ↦ ax + b (mod 26) and decrypt with x ↦ a⁻¹(x − b); a must be coprime to 26. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/diffie-hellman/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/diffie-hellman/Two people agree on a secret g^(ab) mod p while sharing only gᵃ and gᵇ in public. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/rsa/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/rsa/The public key is (n, e) with n = pq; the private key d ≡ e⁻¹ (mod φ(n)); encrypt with c ≡ mᵉ and decrypt with m ≡ cᵈ (mod n). Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/quadratic-residues/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/quadratic-residues/a is a quadratic residue mod p when x² ≡ a has a solution; for an odd prime p, exactly half of the nonzero residues are squares. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/euler-criterion/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/euler-criterion/For p ∤ a, (a/p) is 1 when a is a square mod p and −1 when it is not, and (a/p) ≡ a^((p − 1)/2) (mod p). Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/supplementary-laws/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/supplementary-laws/−1 is a square mod an odd prime p exactly when p ≡ 1 (mod 4), and 2 is a square exactly when p ≡ ±1 (mod 8). Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/quadratic-reciprocity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/quadratic-reciprocity/For odd primes p ≠ q, (p/q) = (q/p), except when p ≡ q ≡ 3 (mod 4), when (p/q) = −(q/p). Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/pythagorean-triples/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/pythagorean-triples/Every primitive triple is (m² − n², 2mn, m² + n²), up to the order of the legs, with m > n > 0, m and n coprime and of opposite parity. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/sums-of-two-squares/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/sums-of-two-squares/n is a sum of two squares exactly when every prime ≡ 3 (mod 4) appears in its factorization to an even power. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/continued-fractions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/continued-fractions/The quotients of the Euclidean algorithm give a/b = [q₀; q₁, q₂, …], and the convergents are better and better approximations. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/irrational-continued-fractions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/irrational-continued-fractions/When d is not a perfect square, √d has a repeating continued fraction, and its convergents are the best rational approximations: no fraction with a smaller denominator comes closer. Number Theory · Math Ladderhttps://teddyrookbook.com/math/number-theory/pell-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/number-theory/pell-equation/For d > 0 not a perfect square, x² − dy² = 1 has a smallest solution with y > 0, found among the convergents of √d, and the powers of x₁ + y₁√d give all the others. Number Theory · Math Ladder