Summary Slides for MATH 342 June 25, 2018

Summary Slides for MATH 342 June 25, 2018 Summary slides based on Elementary Number Theory and its applications by Kenneth Rosen and The Theory of Numbers by Ivan Niven, Herbert Zuckerman, and Hugh Montgomery. 1

Recall the usual notation for the set of natural numbers, integers, rational numbers, real numbers, complex numbers N Z Q R C. Well Ordering Property. Every nonempty set of positive integers has a least element. Definition. A number r R is rational if r = p/q for some integers p, q, q 0. If r is not rational, then it is said to be irrational. Theorem. 2 is irrational. 2

Definition. Let x R. 1. We define [x] to be the greatest integer less than or equal to x, called the greatest integer in x. Note that [x] x < [x] + 1. 2. We define {x} = x [x], called the fractional part of x. Note that 0 {x} < 1. 3

The Pigeonhole Principle. For k N, a function f : {1,..., k + 1} {1,..., k} is not one to one. Theorem. (Dirichlet s Approximation Theorem). If α R and n N, then there exists a, b Z with 1 b n such that bα a < n 1. 4

Definition. If a, b Z with a 0, we say that a divides b, written a b if there is a c Z such that b = ac. We also say in that case that a is a divisor or factor of b and b is a multiple of a. Theorem. If a, b, c Z, a b, b c, then a c. Theorem. If a, b, m, n Z, c a, c b, then c (ma + nb). Theorem. If a b, b 0, then b a. Theorem (The Division Algorithm). If a, b Z with b > 0, then there are unique integers q and r such that a = bq + r with 0 r < b. We call q the quotient, r the remainder, a the dividend, and b the divisor. 5

Definition. The greatest common divisor of two integers a and b, which are not both 0, is the largest integer that divides a and b, and is denoted by (a, b). Definition. The integers a and b are relatively prime if a and b have greatest common divisor (a, b) = 1. Theorem. Let a and b be integers with (a, b) = d. Then (a/d, b/d) = 1. Theorem. Let a, b, c Z. Then (a + cb, b) = (a, b). Definition. If a and b are integers, then a linear combination of a and b is a sum of the form ma + nb, where m, n Z. Theorem. The greatest common divisor of the integers a and b, not both 0, is the least positive integer that is a linear combination of a and b. 6

Corollary. If a and b are relatively prime integers, then there are integers m and n such that ma + nb = 1. Theorem. If a and b are integers, which are not both 0, then the set of linear combinations of a and b is a set of integer multiples of (a, b). Theorem. If a and b are integers, not both 0, then a positive integer d is the greatest common divisor of a and b if and only if d a and d b if c is an integer with c a and c b, then c d. 7

Definition. Let a 1,..., a n be integers, not all 0. The greatest common divisor of these integers is the largest integer that is a divisor of all of the integers in the set. The greatest common divisor of a 1,..., a n is denoted by (a 1,..., a n ). Lemma. If a 1,..., a n are integers, not all 0, then (a 1,..., a n 1, a n ) = (a 1,..., a n 2, (a n 1, a n )). Definition. We say that integers a 1,..., a n are mutually relatively prime if (a 1,..., a n ) = 1. These integers are called pairwise relatively prime if (a i, a j ) = 1 for i j. 8

Theorem (The Euclidean Algorithm). Let r 0 = a and r 1 = b be integers such that a b > 0. If the division algorithm is successively applied to obtain r i = r i+1 q i+1 + r i+2 with 0 < r i+2 < r i+1 for i = 0, 1, 2,..., n 1 and r n+1 = 0, then (a, b) = r n the last nonzero remainder. Lemma. If e and d are integers and e = dq + r, where q and r are integers, then (e, d) = (d, r). 9

Theorem. Let a and b be positive integers. Then (a, b) = x n a+y n b where x n and y n are the n-th terms of the sequences defined recursively by and x 0 = 1, y 0 = 0, x 1 = 0, y 1 = 1 x i = x i 2 q i 1 x i 1, y i = y i 2 q i 1 y i 1. 10

Finite simple continued fraction expansion of a rational number a/b. Set r 0 = a, r 1 = b and apply the Euclidean algorithm: r i = q i r i+1 + r i+2 where 0 < r i+2 < r i+1 for i = 0, 1,..., n 1, and r n+1 = 0. [Note change in indices: q i+2 is now q i and i starts at 0, not 1.] Each step in the Euclidean algorithm expresses: r i /r i+1 = q i + r i+2 /r i+1 = q i + r 1 i+1. r i+2 The finite simple continued fraction expansion of a/b is given by [q 0 ; q 1,..., q n 1 ]. For a real irrational number α, set α 0 = α, and for k 1, let a k = [α k ], and α k = a k + 1 α k+1 (or equivalently, α k+1 = 1 α k a k ). The infinite simple continued fraction expansion of α is given by [a 0 ; a 1, a 2,...]. 11

Definition. A prime is an integer greater than 1 that is divisible by no positive integers other than 1 and itself. Definition. An integer greater than 1 that is not prime is called composite. Lemma. Every integer greater than 1 is a finite product of primes. Theorem. There are infinitely many primes. Theorem. If n is a composite integer, then n has a prime factor not exceeding n. Sieve of Erathosthenes 12

Theorem (The Fundamental Theorem of Arithmetic). Every positive integer greater than 1 can be written uniquely as a product of primes, with the prime factors in the product written in nondecreasing order. Lemma. If a, b, c Z and (a, b) = 1 and a bc, then a c. Lemma. If p a 1... a n where p is a prime and a 1,..., a n Z, then p a i for some i. Lemma. If (a, m) = 1 and (b, m) = 1, then (ab, m) = 1. Definition. The least common multiple of two nonzero integers a and b is the smallest positive integer that is divisible by a and b, and is denoted by [a, b]. Theorem. If a and b are positive integers, then [a, b] = ab/(a, b). 13

Parity of an integer: An integer is odd if and only if it is of the form 2k + 1 for some k Z. An integer is even if and only if it is of the form 2k, for some k Z. 14

Theorem. Let α R be a root of a monic polynomial with coefficients in Z. Then α Z or α is irrational. Theorem. Let a and b be non-zero integers with d = (a, b). The equation ax + by = c has no integral solutions if d c. If d c, then there are infinitely many integral solutions. Moreover, if x = x 0, y = y 0 is a particular solution of the equation, then all solutions are given by x = x 0 + (b/d)n, y = y 0 (a/d)n. 15

Theorem. Let n 2. If a 1,..., a n are nonzero integers, then the equation a 1 x 1 +...+a n x n = c has an integral solution if and only if d = (a 1,..., a n ) divides c. Furthermore, when there is a solution, there are infinitely many solutions. Remark: Section 5.1 and 5.2 gives a general method for solving a system of linear diophantine equations in several variables. 16

Definition. Let m be a positive integer. If a and b are integers, we say that a is congruent to b modulo m if m (a b), and write a b (mod m). Theorem. If a and b are integers, then a b (mod m) if and only if there is an integer k such that a = b + km. Theorem. Let m N. Congruences modulo m satisfy the following properties. a a (mod m) a b (mod m) implies b a (mod m). a b (mod m), b c (mod m) implies a c (mod m). 17

Theorem. If a, b, c, m Z, m > 0, and a b (mod m) then a + c b + c (mod m) a c b c (mod m) ac bc (mod m). 18

Theorem. Let a, b, c, d, e, f, m Z, m > 0, (, m) = 1 where = ad bc. Then the system of congruences ax + by e (mod m) cd + dy f (mod m) has a unique solution modulo m. 19

Theorem. If a, b, c, m Z, m > 0, d = (c, m), and ac bc (mod m), then a b (mod m/d). Corollary. If a, b, c, m Z, m > 0, (c, m) = 1, and ac bc (mod m), then a b (mod m). Definition. A complete residue system modulo m is a set of integers such that every integer is congruent modulo m to exactly one integer in this set. Lemma. A set of m incongruent integers modulo m forms a complete set of residues modulo m. Theorem. If r 1,..., r m is a complete residue system modulo m, and if (a, m) = 1, then ar 1 + b, ar 2 + b,..., ar m + b is a complete system of residues modulo m for any integer b. Theorem. If a, b, k, m Z, k > 0, m > 0, and a b (mod m), then a k b k (mod m). 20

Theorem. If a b (mod m 1 ),..., a b (mod m k ), where a, b, m 1, m 2,..., m k Z then a b (mod [m 1, m 2,..., m k ]). Corollary. If a b (mod m 1 ),..., a b (mod m k ), where a and b are integers, and m 1,..., m k are pairwise relatively prime integers, then a b (mod m 1... m k ).

Theorem. Let a, b, m Z, m > 0 and (a, m) = d. If d b, then ax b (mod m) has no solutions. If d b, then ax b (mod m) has exactly d incongruent solutions modulo m. Corollary. If a and m are relatively prime integers with m > 0 and b an integer, then the linear congruence ax b (mod m) has a unique solution. Definition. Given an integer a with (a, m) = 1, a solution of ax 1 (mod m) is called an inverse of a modulo m. Theorem. Let p be prime. The positive integer a is its own inverse modulo p if and only if a 1 (mod p) or a 1 (mod p). 21

Digression into abstract algebra (not part of course material) A group G is a set with a binary composition law such that 1. For all a, b, c G, (ab)c = a(bc). 2. There is an element e G such that ae = ea = a for all a G. 3. For each a G, there is an b G such that ab = ba = e. The identity element e is unique. The inverse element b of a is unique, denoted a 1. 22

A group G is said to be commutative or abelian if xy = yx for all x, y G. Sometimes we denote compositions additively as the sum x + y when the group G is abelian. In that case, we denote the identity element as zero 0 and inverses as negatives x. A group G is said to be commutative or abelian if xy = yx for all x, y G. A field K is a non-empty set with two compositions laws, addition and multiplication such that 1. K is an abelian group under addition (0 is additive identity, negation is additive inverse)

2. 0x = x0 = 0 for all x K 3. K = K {0} is an abelian group under multiplication (1 is multiplicative identity, reciprocal is multiplicative inverse) 4. x(y + z) = xy + yz for all x, y, z K It is automatic that 0 1, (x + y)z = xz + yz for all x, y, z K, 1x = x = x1 for all x K. A ring R is a non-empty set with two compositions laws, addition and multiplication such that 1. R is an abelian group under addition 2. (xy)z = x(yz) for all x, y, z R

3. x(y + z) = xy + xz for all x, y, z R 4. (x + y)z = xz + yz for all x, y, z R The identity element of addition is denoted by zero 0. It follows that 0x = 0 = x0 for all x R. The properties for congruences imply: Z/mZ is a ring Z/pZ is a field for p prime Z/mZ under + is an abelian group (Z/mZ) under is an abelian group

Theorem (Wilson s Theorem). then (p 1)! 1 (mod p). If p is prime, Theorem. If n 2 is an integer such that (n 1)! 1 (mod n), then n is prime. Theorem (Fermat s Little Theorem). If p is prime and a is an integer with p a, then a p 1 1 (mod p). Theorem. If p is prime then a p a (mod p). Theorem. If p is prime and a is an integer such that p a, then a p 2 is an inverse of a modulo p. Corollary. If a and b are positive integers and p is prime with p a, then the solutions of the linear congruence ax b (mod p) are the integers x such that x = a p 2 b (mod p). 23

Definition. Let n be a positive integer. The Euler φ function φ(n) is defined to be the number of positive integers not exceeding n that are relatively prime to n. Definition. A reduced residue system modulo n is a set of φ(n) integers such that each element of the set is relatively prime to n, and no two different elements of the set are congruent modulo n. Theorem. If r 1,..., r φ(n) is a reduced residue system modulo n, and if a is a positive integer with (a, n) = 1, then the set ar 1,..., ar φ(n) is also a reduced residue system modulo n. Theorem (Euler s Theorem). If m is a positive integer and a is an integer with (a, m) = 1, then a φ(m) 1 (mod m). 24

Theorem (The Chinese Remainder Theorem). Let m 1,..., m r be pairwise relatively prime positive integers. Then the system of congruences x a 1 (mod m 1 ) x a 2 (mod m 2 )...... x a r (mod m r ) has a unique solution modulo M = m 1... m r. 25

Theorem. Let b, m, n are positive integers such that b < m. Then the least positive residue of b N modulo m can be computed using bit operations. O((log 2 m) 2 log 2 N) 26

Definition. Let b be a positive integer. If n is a composite positive integer and b n 1 1 (mod n), then n is called a pseudoprime to the base b. Lemma. If d and n are positive integers such that d n, then 2 d 1 divides 2 n 1. Theorem. There are infinitely many pseudoprimes to the base 2. Definition. A composite number n that satisfies b n 1 1 (mod n) for all positive integers b with (b, n) = 1 is called a Carmichael number or absolute pseudoprime. Theorem. If n = q 1... q k where q j are distinct prime numbers that satisfy (q j 1) (n 1) for all j and k > 2, then n is a Carmichael number. 27

Primitive Factorization Methods 1. Trial division. 2. Fermat factorization method. 3. Pollard p 1 factorization method. (see Screencast 2: Primitive Factorization Methods) 28

Definition. An arithmetic function is a function that is defined for all positive integers. Definition. An arithmetic function f is called multiplicative if f(mn) = f(m)f(n) whenever m and n are relatively prime positive integers. It is called completely multiplicative if f(mn) = f(m)f(n) for all positive integers m and n. Theorem. If f is a multiplicative function and if n = p a 1 1... pa s s is the prime power factorization of n, then f(n) = f(p a 1 1 )... f(pa s s ). Theorem. If p is prime, then φ(p) = p 1. Conversely, if p is a positive integer with φ(p) = p 1, then p is prime. Theorem. Let p be a prime and a a positive integer. Then φ(p a ) = p a p a 1. Theorem. Let m and n be relatively prime positive integers. Then φ(mn) = φ(m)φ(n). 29

Theorem. Let n = p a 1 1... pa k k be the prime power factorization of the positive integer n. Then ) ) φ(n) = n (1 1p1... (1 1pk. Theorem. Let n be a positive integer greater than 2. Then φ(n) is even. Theorem. Let n be a positive integer. Then d n φ(d) = n. Definition. The sum of divisors function, denoted by σ, is defined by setting σ(n) equal to the sum of all positive divisors of n. Definition. The number of divisors function, denoted by τ, is defined by setting τ(n) equal to the number of positive divisors of n. 30

Theorem. If f is a multiplicative function, then the summatory function of f, namely F (n) = d n f(d) is also multiplicative. Corollary. The sum of divisors function σ and the number of divisors function τ are multiplicative functions. Lemma. Let p be prime and a a positive integer. Then and σ(p a ) = 1 + p + p 2 +... + p a = pa+1 1 p 1 τ(p a ) = a + 1. 31

Theorem. Let the positive integer n have prime factorization n = p a 1 1... pa s s. Then and σ(n) = pa 1+1 1 1 p 1 1... pas+1 s 1 p s 1, τ(n) = (a 1 + 1)... (a s + 1). Definition. If n is a positive integer and σ(n) = 2n, then n is called a perfect number. Theorem. The positive integer n is an even perfect number if and only if n = 2 m 1 (2 m 1) and 2 m 1 is prime. Theorem. If m is a positive integer and 2 m 1 is prime, then m must be prime. Definition. If m is a positive integer, then M m = 2 m 1 is called the mth Mersenne number. If p is prime and M p = 2 p 1 is prime, then M p is called a Mersenne prime. 32

Theorem. If p is an odd prime, then any divisor of the Mersenne number M p = 2 p 1 is of the form 2kp + 1, where k is a positive integer.

Definition. A positive integer n is squarefree if n > 1 and there is no prime p such that p 2 n. The Möbius function µ(n) is de- Definition. fined by µ(n) = 1 if n = 1 ( 1) r if n = p 1... p r is squarefree 0 otherwise Lemma. Let m and n are relatively prime positive integers. Then if d is a positive divisor of mn, there is a unique pair of positive divisors d 1 of m and d 2 of n such that d = d 1 d 2. Conversely, if d 1 and d 2 are positive divisors of m and n, respectively, then d = d 1 d 2 is a positive divisor of mn. Theorem. The Möbius function µ(n) is a multiplicative function. Theorem. The summatory function of the Möbius function at the integer n, F (n) = d n µ(d) satisfies F (n) = 1 if n = 1 and F (n) = 0 if n > 1. 33.

Theorem (The Möbius Inversion Formula). Suppose that f is an arithmetic function and that F is the summatory function of f so that F (n) = d n f(d). Then for all positive integers n, f(n) = d n µ(d)f (n/d). Theorem. Let f be an arithmetic function with summatory function F. Then if F is a multiplicative, f is also multiplicative. 34

Definition. Let a and n be relatively prime integers. Then the least positive integer x such that a x 1 (mod n) is called the order of a modulo n, denoted ord n a. Theorem. If a and n are relatively prime integers with n > 0, then the positive integer x is a solution of the congruence a x 1 (mod n) if and only if ord n a x. Corollary. If a and n are relatively prime integers with n > 0, then ord n a φ(n). Theorem. If a and n are relatively prime integers with n > 0, then a i a j (mod n), where i and j are nonnegative integers, if and only if i j (mod ord n a). Definition. If r and n are relatively prime integers with n > 0 and if ord n r = φ(n), then r is a primitive root modulo n. 35

Theorem. If r and n are relatively prime positive integers and if r is a primitive root modulo n, then the integers r 1, r 2,..., r φ(n) form a reduced residue set modulo n. Theorem. If ord n a = t and if u is a positive integer, then ord n (a u ) = t/(t, u). Corollary. Let r be a primitive root modulo n, where n is an integer, n > 1. Then r u is a primitive root modulo n if and only if (u, φ(n)) = 1. Theorem. If the positive integer n has a primitive root, then it has a total of φ(φ(n)) incongruent primitive roots. 36

Theorem (Lagrange s Theorem). Let f Z[x] be a polynomial of degree n 1 with leading coefficient a n not divisible by p a prime. Then f has at most n incongruent roots modulo p. Theorem. Let p be prime and let d be a divisor of p 1. Then the polynomial x d 1 has exactly d incongruent roots modulo p. Lemma. Let p be a prime and let d be a positive divisor of p 1. Then the number of positive integers less than p of order d modulo p does not exceed φ(d). Theorem. Let p be a prime and let d be a positive divisor of p 1. Then the number of incongruent integers of order d modulo p is equal to φ(d). Corollary. Every prime has a primitive root. Artin s Conjecture. The integer a is a primitive root of infinitely many primes if a ±1 and a is not a perfect square. 37

Theorem. If p is an odd prime with primitive root r, then either r or r + p is a primitive root modulo p 2. Theorem. Let p be an odd prime. Then p k has a primitive root for all positive integers k. Moreover, if r is a primitive root modulo p 2, then r is a primitive root modulo p k for all positive integers k. Theorem. If a is an odd integer, and if k 3 is an integer, then a φ(2k )/2 = a 2k 2 1 (mod 2 k ). Theorem. If n is a positive integer that is not a prime power or twice a prime power, then n does not have a primitive root. Theorem. If p is an odd prime and t is a positive integer, then 2p t possesses a primitive 38

root. In fact, if r is a primitive root modulo p t, then if r is odd, it is also a primitive root modulo 2p t ; whereas if r is even, r + p t is a primitive root modulo 2p t. Theorem. The positive integer n > 1 possesses a primitive root if and only if n = 2, 4, p t, 2p t where p is an odd prime and t N.

Definition. Let m be a positive integer with primitive root r. If a is a positive integer with (a, m) = 1, then the unique integer x with 1 x φ(m) and r x a (mod m) is called the index or discrete logarithm of a to the base r modulo m, denote ind r a. Theorem. Let m be a positive integer with primitive root r and let a and b be integers relatively prime to m. Then ind r 1 = 0 (mod φ(m)) ind r ab ind r a + ind r b (mod φ(m)) ind r a k k ind r a (mod φ(m)). Theorem. Let m be a positive integer with a primitive root. If k is a positive integer and 39

a is an integer relatively prime to m, then the congruence x k a (mod m) has a solution if and only if a φ(m)/d 1 (mod m) where d = (k, φ(m)).

Theorem (Dirichlet s Theorem on Primes in Arithmetic Progressions). Suppose that a, b N are not divisible by the same prime. Then the arithmetic progression an+b, n = 1, 2, 3,... contains infinitely many primes. Definition. Let x R. Define π(x) to be the number of prime numbers x. Theorem (The Prime Number Theorem). The ratio of π(x) to x/ log x approaches 1 as x grows without bound. Corollary. Let p n denote the nth prime, when n N. Then p n n log n. Theorem. For any positive integer n, there are at least n consecutive composite positive integers. Theorem (Bertrand s Postulate) For every positive integer n > 1, there is a prime p such that n < p < 2n. 40

Theorem (Chebychev) Let a < a 0 = 1 3 log 2, b > b 0 = 3 2 a 0. Then there exists an x 0 such that for all x x 0. a x log x < π(x) < b x log x 41

The Twin Prime Conjecture. There are infinitely many pairs of primes p and p + 2. Goldbach s Conjecture. Every even positive integer greater than 2 is the sum of two primes. The n 2 + 1 Conjecture. There are infinitely many primes of the form n 2 + 1, where n is a positive integer. 42

Definition. If m is a positive integer, we say that the integer a is a quadratic residue of m if (a, m) = 1 and the congruence x 2 a (mod m) has a solution. If (a, m) = 1 and the congruence x 2 a (mod m) has no solution, we say that a is a quadratic nonresidue of m. Lemma. Let p be an odd prime and a and integer not divisible by p. Then, the congruence x 2 a (mod p) has either no solutions or exactly two incongruent solutions modulo p. Theorem. If p is an odd prime, then there are exactly (p 1)/2 quadratic residues of p and (p 1)/2 quadratic nonresidues of p among the integers 1, 2,..., p 1. 43

Theorem. Let p be a prime and let r be a primitive root of p. If a is an integer not divisible by p, then a is a quadratic residue of p if ind r a is even and a is a quadratic nonresidue of p if ind r a is odd. Definition. Let p be an odd prime and a be an integer no divisible by p. The Legendre symbol ( ap ) is defined by ( ) a p = 1 if a is a quadratic residue of p 1 if a is a quadratic nonresidue of p Theorem (Euler s Criterion). Let p be an odd prime and let a be a positive integer not divisible by p. Then ( ) a p a (p 1)/2 (mod p). 44

Theorem. Let p be an odd prime and a and b be integers not divisible by p. Then if a b (mod p), then ( a p ) = ( bp ) ( a p ) ( bp ) = ( ab p ) ( ) a 2 p = 1 Theorem. If p is an odd prime, then ( 1 p ) = 1 if p 1 (mod 4) 1 if p 1 (mod 4). 45

Lemma (Gauss Lemma). Let p be an odd prime and a an integer with (a, p) = 1. If s is the number of least positive residues of the integers a, 2a, 3a,..., p 1 a that are greater than p/2, then ( a p ) = ( 1) s. 2 Theorem. If p is an odd prime, then ( 2 p ) = ( 1) (p2 1)/8. Theorem (The Law of Quadratic Reciprocity). Let p and q be distinct odd primes. Then ( ) ( ) p q q p = ( 1) p 1 q 1 2 2. Lemma. If p is an odd prime and a is an odd integer not divisible by p, then ( a p ) = ( 1) T (a,p) where T (a, p) = (p 1)/2 j=1 [ja/p]. 46

Definition. Let n be an odd positive integer with prime factorization n = p t 1 1... p t m and let a be an integer relatively prime to n. Then, the Jacobi symbol ( ) a n is defined by ( ) a n = ( a p 1 ) t1... ( ) tm a. p m Theorem. Let n be an odd positive integer and let a and b be integers relatively prime to n. Then if a b (mod n), then ( a n ) = ( bn ) ( ab n ) = ( an ) ( bn ) ( 1 n ) = ( 1) (n 1)/2 ( 2 n ) = ( 1) (n 2 1)/8. 47

Theorem (The Reciprocity Law for Jacobi Symbols). Let n and m be relatively prime odd positive integers. Then ( ) ( ) n mn m 1 m = ( 1) 2 n 1 2. Let R 0 = a and R 1 = b. Using the division algorithm and factoring out the highest power of two dividing the remainders, we obtain R 1 = R 2 q 2 + 2 s 2R 3 R 2 = R 3 q 3 + 2 s 3R 4... R n 2 = R n 1 q n 1 + 2 s n 1 1, where s j is a nonnegative integer and R j is an odd positive integer less than R j 1 for j = 2, 3,..., n 1. 48

Theorem. Let a and b be positive integers with a > b. Then ( ) a b = ( 1) w R w = s 1 2 1 Rn 1 2 1 +... + s 1 n 1 8 8 + R 1 1R 2 1 +... + R n 2 1R n 1 1. 2 2 2 2 Remark: see examples in class and textbook on how to calculate Jacobi symbols in practice. Corollary. Let a and b be relatively prime positive integers with a > b. Then the Jacobi symbol ( a b ) can be evaluated using O((log2 b) 3 ) bit operations. 49

Theorem. If p is a prime, not of the form 4k + 3, then there are integers x and y such that x 2 + y 2 = p. Theorem. Every solution to x 2 + y 2 = z 2 with x, y, z positive integers, (x, y, z) = 1, x odd, y even is given by x = m 2 n 2, y = 2mn, z = m 2 +n 2, where (m, n) = 1 and m n (mod 2). Theorem. The equation x 4 + y 4 = z 2 has no solutions in non-zero integers x, y, z. 50

Theorem. If m and n are both sums of two squares, then mn is also the sum of two squares. Theorem. If p is a prime, not of the form 4k + 3, then there are integers x and y such that x 2 + y 2 = p. Theorem. The positive integer n is the sum of two squares if and only if each prime factor of n of the form 4k + 3 occurs to an even power in the prime factorization of n. Theorem. The positive integer n is the sum of two coprime squares if and only if each prime factor of n is of the form 4k + 1 or is the prime 2 which occurs to at most a first power. Theorem. If m and n are positive integers that are each the sum of four squares, then mn is also the some of four squares. 51

Theorem. Let p be a prime. Then p is the sum of the squares of four integers. Theorem. Every positive integer is the sum of the squares of four integers.

For material on ad hoc methods for solving diophantine equations and Hasse s theorem, see the textbook. Theorem (Liouville). Let α R be a root of an irreducible polynomial in Z[X] of degree d > 1. Then there exists a constant C = C(α) > 0 such that α p/q > C(α)/q d for all rational numbers p/q. 52

Theorem. The real number α = i=1 1 10 i! is transcendental over Q.