MAU23101 Introduction to number theory 4 - Sums of squares Nicolas Mascot [email protected] Module web page Michaelmas 2020–2021 Version: November 2, 2020 Nicolas Mascot Introduction to number theory Theorem Q vj An integer n j p N is a sum of 2 squares iff. vj is even Æ j 2 whenever pj 1 mod 4. ´ ¡ Theorem (Legendre) An integer n N is a sum of 3 squares iff. it is not of the form a 2 4 (8b 7), a,b Z 0. Å 2 Ê Theorem (Lagrange) Every n N is a sum of 4 squares. 2 Main goal of this chapter Theorem (Obvious) Q vj An integer n j p N is a square iff. vj is even for all j. Æ j 2 Example 2020 22511011 is not a square. Æ Nicolas Mascot Introduction to number theory Theorem (Legendre) An integer n N is a sum of 3 squares iff. it is not of the form a 2 4 (8b 7), a,b Z 0. Å 2 Ê Theorem (Lagrange) Every n N is a sum of 4 squares. 2 Main goal of this chapter Theorem Q vj An integer n j p N is a sum of 2 squares iff. vj is even Æ j 2 whenever pj 1 mod 4. ´ ¡ Example 2019 316731 is not a sum of 2 squares. 3 2019Æ 326731 362 692. 2020£ 22Æ511011 Æ242 Å382. 32 3Æ2 02. Æ Å Æ Å Nicolas Mascot Introduction to number theory Theorem (Lagrange) Every n N is a sum of 4 squares. 2 Main goal of this chapter Theorem Q vj An integer n j p N is a sum of 2 squares iff. vj is even Æ j 2 whenever pj 1 mod 4. ´ ¡ Theorem (Legendre) An integer n N is a sum of 3 squares iff. it is not of the form a 2 4 (8b 7), a,b Z 0. Å 2 Ê So n is not a sum of 3 squares iff. v2(n) is even and n 1 mod 8. 2v2(n) ´ ¡ Example 60 2215 is not a sum of 3 squares. 30 Æ 21 15 52 22 12. 44 22 11 62 22 22. Æ £ Æ Å Å Æ £ Æ Å Å Nicolas Mascot Introduction to number theory Main goal of this chapter Theorem Q vj An integer n j p N is a sum of 2 squares iff. vj is even Æ j 2 whenever pj 1 mod 4. ´ ¡ Theorem (Legendre) An integer n N is a sum of 3 squares iff. it is not of the form a 2 4 (8b 7), a,b Z 0. Å 2 Ê Theorem (Lagrange) Every n N is a sum of 4 squares. 2 Example 60 62 42 22 22. Æ Å Å Å Nicolas Mascot Introduction to number theory Waring’s problem (not examinable) Theorem (Hilbert, 1909) For each k N, there exists m N such that every n N is the sum of m k2th powers. 2 2 For each k, the smallest possible m is denoted by g(k). Theorem g(2) 4 (Lagrange, 1770) Æ g(3) 9 (Wieferich - Kempner, 1910) Æ » g(4) 19 (Balasubramanian - Dress - Deshouillers, 1986) Æ g(5) 37 (Chen, 1964) Æ ¢¢¢ Nicolas Mascot Introduction to number theory Gaussian integers Nicolas Mascot Introduction to number theory Gaussian integers Definition The set of Gaussian integers is Z[i] {a bi a,b Z} C, Æ Å j 2 ½ where i C is such that i 2 1. 2 Æ¡ Proposition Z[i] is a ring: whenever ®,¯ Z[i], we also have 2 ® ¯, ® ¯, ®¯ Z[i]. Å ¡ 2 Proof. (a bi)(c di) (ac bd) (ad bc)i. Å Å Æ ¡ Å Å Nicolas Mascot Introduction to number theory Gaussian integers Definition The set of Gaussian integers is Z[i] {a bi a,b Z} C, Æ Å j 2 ½ where i C is such that i 2 1. 2 Æ¡ Proposition Z[i] is a ring: whenever ®,¯ Z[i], we also have 2 ® ¯, ® ¯, ®¯ Z[i]. Å ¡ 2 Remark Z[i] {P(i) P(x) Z[x]}, whence the notation Z[i]. Æ j 2 Nicolas Mascot Introduction to number theory The norm Definition The norm of ® a bi Z[i] is Æ Å 2 N(®) ®® a2 b2. Æ Æ Å Remark N(®) 0, with equality only if ® 0. Ê Æ If n Z Z[i], then N(n) n2. 2 ½ Æ Proposition For all ®,¯ Z[i], N(®¯) N(®)N(¯). 2 Æ Lemma An integer n N is a sum of 2 squares iff. it is the norm of a Gaussian integer.2 Nicolas Mascot Introduction to number theory Units Definition A Gaussian integer ® Z[i] is a unit if it is invertible in Z[i], 2 meaning there exists ¯ Z[i] such that ®¯ 1. The set of 2 Æ units of Z[i] is denoted by Z[i]£. Proposition Let ® Z[i]. Then ® is a unit iff. N(®) 1. 2 Æ Proof. If ®¯ 1, then 1 N(1) N(®¯) N(®)N(¯). Æ Æ Æ Æ Conversely, if N(®) 1, then ®¯ 1 for ¯ ® Z[i]. Æ Æ Æ 2 Nicolas Mascot Introduction to number theory Units Definition A Gaussian integer ® Z[i] is a unit if it is invertible in Z[i], 2 meaning there exists ¯ Z[i] such that ®¯ 1. The set of 2 Æ units of Z[i] is denoted by Z[i]£. Proposition Let ® Z[i]. Then ® is a unit iff. N(®) 1. 2 Æ Corollary Z[i]£ {1, 1,i, i}. Æ ¡ ¡ Remark We could say that in Z, the units are 1 and 1; hence the term “unit”. ¡ Nicolas Mascot Introduction to number theory Arithmetic with the Gaussian integers Nicolas Mascot Introduction to number theory Euclidean division Theorem Let ®,¯ Z[i] with ¯ 0. There exists γ,½ Z[i] such that 2 6Æ 2 ® βγ ½ and N(½) N(¯). Æ Å Ç Nicolas Mascot Introduction to number theory Euclidean division Proof. Compute ®/¯ x yi C. Let m,n Z such that Æ Å 2 2 1 1 x m and y n , j ¡ j É 2 j ¡ j É 2 and set γ m ni, ½ ® ¯γ. Then γ,½ Z[i], and Æ Å Æ ¡ 2 ® ¯γ ½. Æ Å Extend the norm to all of C by N(®) ®®. Then Æ N(½) N(® ¯γ) µ® ¶ ¡ N γ N((x yi) (m ni)) N(¯) Æ N(¯) Æ ¯ ¡ Æ Å ¡ Å µ1¶2 µ1¶2 1 (x m)2 (y n)2 , Æ ¡ Å ¡ É 2 Å 2 Æ 2 so N(½) 1 N(¯) N(¯). É 2 Ç Nicolas Mascot Introduction to number theory Euclidean division Theorem Let ®,¯ Z[i] with ¯ 0. There exists γ,½ Z[i] such that 2 6Æ 2 ® βγ ½ and N(½) N(¯). Æ Å Ç Example Let ® 8 i, ¯ 2 3i. Then ƮŠ8 Æi Å (8 i)(2 3i) 19 22 Å Å ¡ i 1 2i, ¯ Æ 2 3i Æ (2 3i)(2 3i) Æ 13 ¡ 13 ¼ ¡ Å Å ¡ so we set γ 1 2i and ½ ® ¯γ 2i. Æ ¡ Æ ¡ Æ We can check that N(½) 4 N(¯) 13. Æ Ç Æ Remark In general, the pair (γ,½) is not unique. But it will not matter for what we have in mind! Nicolas Mascot Introduction to number theory Consequences of Euclidean division: gcd Definition Let ®,¯ Z[i]. We say that ® ¯ if there exists γ Z[i] such 2 j 2 that ¯ αγ. Æ Lemma (Important) For all ® Z[i], we have ® N(®). 2 j If ® ¯ in Z[i] , then N(®) N(¯) in Z. j j Nicolas Mascot Introduction to number theory Consequences of Euclidean division: gcd Definition We say that ®,¯ Z[i] are associate if ® ¯ and ¯ ®. 2 j j Lemma ®,¯ are associate ¯ υα for some υ Z[i]£. () Æ 2 Proof. 1 : If ¯ υα, then ® ¯, and also ® υ¡ ¯ so ¯ ®. ( Æ j Æ j : ¯ »® and ® ´¯, so ® (1 »´)®. ) Æ Æ Æ ¡ If ® 0 then »´ 1 so »,´ Z[i]£. 6Æ Æ 2 If ® 0 then ¯ »® 0 so also OK. Æ Æ Æ Nicolas Mascot Introduction to number theory Consequences of Euclidean division: gcd Definition Let ®,¯,γ Z[i]. We say that γ is a gcd of ®, ¯ if 2 for all ± Z[i], ± γ ± ® and ± ¯. 2 j () j j Alternatively, a gcd is a common divisor whose norm is as large as possible. Theorem Gcd’s exist, can be found by the Euclidean algorithm, and are unique up to multiplication by units. Nicolas Mascot Introduction to number theory Consequences of Euclidean division: gcd Theorem Gcd’s exist, can be found by the Euclidean algorithm, and are unique up to multiplication by units. Proof. If ® ¯γ ½, then Div(®,¯) Div(¯,½) Gcd’s exist and canÆ be foundÅ by Euclidean algorithm.Æ Uniqueness: suppose ®,¯ are not both 0, and let γ,γ0 be two gcd’s. Then γ γ0 and γ0 γ. j j Corollary Given ®,¯, the elements of Z[i] of the form ®» ¯´ are Å exactly the multiples of gcd(®,¯). Gauss’s lemma: if ® ¯γ and gcd(®,¯) 1, then ® γ. j Æ j Nicolas Mascot Introduction to number theory Consequences of Euclidean division: factorisation Definition (Gaussian primes) An element ¼ Z[i] is irreducible if ¼ Z[i]£ and whenever 2 62 ¼ ®¯, then one of ®,¯ is a unit. Æ Example If N(®) is a prime number, then ® is irreducible. Indeed, if ® ¯γ, then N(®) N(¯)N(γ). Æ Æ "The converse is not true! Nicolas Mascot Introduction to number theory Consequences of Euclidean division: factorisation Theorem Every nonzero ® Z[i] may be factored as 2 ® υπ1 ¼r Æ ¢¢¢ with υ Z[i]£ and the ¼j irreducible. 2 If ® υ0¼0 ¼0 , then r s and each ¼0 is associate to a ¼k .