7/8/2011 Lagrange’s Four Square Theorem: Any natural number 푁 can be represented as 푁 = 푎2 + 푏2 + 푐2 + 푑2 where 푎, 푏, 푐, and 푑 are integers. Erin Compaan and Cynthia Wu SPWM 2011 Our goal is to prove this theorem using Hurwitz Quaternions. Diophantus – ca. 200 A.D. Denoted by ℍ Bachet – 1621 Members of a non-commutative division algebra th Fermat – 17 c. Lagrange – 1770 Form of quaternions: 푎 + 푏푖 + 푐푗 + 푑푘 where 푎, 푏, 푐, and 푑 are real numbers Fundamental formula of quaternion algebra: 푖2 = 푗2 = 푘2 = 푖푗푘 = −1 1 7/8/2011 Not that much of a difference! A Hurwitz quaternion 훼 is prime if it divisible only 1 1 by the quaternions ±1, ±푖, ±푗, ±푘, and ± ± 푖 ± 1 2 2 푯 = *푎 + 푏푖 + 푐푗 + 푑푘 휖ℍ ∶ 푎, 푏, 푐, 푑 휖 ℤ or 푎, 푏, 푐, 푑 휖 ℤ + + 1 1 2 푗 ± 푘, and multiples of 훼 with these. 2 2 So now 푎, 푏, 푐, and 푑 are either all integers or all half integers A Hurwitz quaternion 훽 divides 훼 if there exists a Hurwitz quaternion 휑 such that 훼 = 훽휑 or 훼 = 휑훽. Half integers: all numbers that are half of an odd 1 integer – the set ℤ + . 2 ◦ EG: 7/2, -13/2, 8.5 A Lipschitz quaternion is a quaternion of the form 푎 + 푏푖 + 푐푗 + 푑푘, with a, b, c, d 휖 ℤ. Any natural number 푁 can be represented as 2 2 2 2 ◦ E.g. 1 + 7푖 − 83푗 + 12푘. 푁 = 푎 + 푏 + 푐 + 푑 where 푎, 푏, 푐, and 푑 are integers. Prove this using Hurwitz Quaternions: 푯 = *푎 + 푏푖 + 푐푗 + 푑푘 휖ℍ ∶ 푎, 푏, 푐, 푑 휖 ℤ or 푎, 푏, 푐, 푑 휖 ℤ+ 2 7/8/2011 If two numbers can be written as a sum of four integer If 푝 is a prime 푝 = 2푛 + 1, 푛 휖 ℕ, then there are squares, then so can their product. 2 2 2 2 2 2 푙, 푚 휖 ℤ such that 푝 divides 1 + 푙 + 푚 . Proof: Suppose that 푢 = 푎 + 푏 + 푐 + 푑 and 2 2 2 2 푣 = 푤 + 푥 + 푦 + 푧 . If a Hurwitz prime divides a product of 2 2 Hurwitz quaternions 훼훽, then the prime Then 푢 = 푎 + 푏푖 + 푐푗 + 푑푘 = 훼 and 푣 = 푤 + 푥푖 + 푦푗 + 푧푘 2 = 훽 2, for some Lipschitz divides 훼 or 훽. quaternions 훼 and 훽. Then 2 2 푢푣 = 훼 훽 = 훼훽 = 퐴 + 퐵푖 + 퐶푗 + 퐷푘 2= 퐴2 + 퐵2 + 퐶2 + 퐷2 for some 퐴, 퐵, 퐶, 퐷 휖 ℤ. Suppose 푝 is an odd prime which has a non-trivial Hurwitz factorization 푝 = (푎 + 푏푖 + 푐푗 + 푑푘)훼. Base Cases Conjugating: 푝 = 푝 = 훼 (푎 − 푏푖 − 푐푗 − 푑푘). 1 = 12 + 02 + 02 + 02 2 2 2 2 Multiplying the equations: 2 = 1 + 1 + 0 + 0 푝2 = 푎 + 푏푖 + 푐푗 + 푑푘 훼훼 푎 − 푏푖 − 푐푗 − 푑푘 2 2 2 2 2 = 푎 + 푏 + 푐 + 푑 훼 2 Since 푝 is prime, the factors of 푝 must both be 푝. Thus 푝 = 푎2 + 푏2 + 푐2 + 푑2. If 푎, 푏, 푐, and 푑 are integers , we’re done. If not, we can still show that p is a sum of four integer squares. 3 7/8/2011 Now let 푝 be an odd prime. Then there exist integers Since 푝 is not a Hurwitz prime, we can apply our 푙 and 푚 such that 푝 divides 1 + 푙2 + 푚2. previous conclusion and say that 푝 is a sum of four Then 푝 divides (1 + 푙푖 + 푚푗)(1 − 푙푖 − 푚푗). By the integer squares. previously stated lemma, if 푝 were a Hurwitz prime, We now have that 1, 2, and all odd primes can be it must divide one of these factors. written as a sum of four squares. 1 푙 푚 But this would imply that + 푖 + 푗 or By the Four Squares identity, every natural number 푝 푝 푝 1 푙 푚 can be written as a sum of four squares. − 푖 − 푗 is a Hurwitz integer, a contradiction. 푝 푝 푝 Thus 푝 is not a Hurwitz prime. Fermat’s Two Square Theorem: If a prime 푝 is of the form 4푛 + 1 2 2 Suppose p is a prime of the form 4푛 + 1 for some for some 푛 휖 ℕ, then 푝 = 푎 + 푏 for some 푎, 푏 휖 ℤ. 푛 휖 ℕ. 2 Gaussian integers: Complex numbers with integer coefficients. Then p divides 1 + 푚 = (1 + 푚푖)(1 − 푚푖) for some 푚 휖 ℕ. 2 Gaussian integer prime: A Gaussian integer z which is divisible Since p divides neither factor of 1 + 푚 , 푝 is not a only by ±1 or ± 푖, or products of z with these. Gaussian prime. Then p has a nontrivial factorization in the Gaussian Lemma: For any prime 푝 of the form 4푛 + 1, 푛 휖 ℕ, there exists an integer 푚 such that 푝 divides 1 + 푚2. integers 푝 = (푎 + 푏푖)(푥 + 푦푖). Lemma: If a Gaussian integer prime 푝 divides 훼훽 for some Gaussian integers 훼 and 훽, then 푝 divides 훼 or 푝 divides 훽. 4 7/8/2011 Conjugating and multiplying equations 2 푝 = 푎 + 푏푖 푥 + 푦푖 푎 + 푏푖 푥 + 푦푖 Various contributions = (푎 + 푏푖)(푎 − 푏푖)(푥 + 푦푖)(푥 − 푦푖) Calculus of Variations, 2 2 2 2 Lagrange Multipliers, = 푎 + 푏 푥 + 푦 . PDE’s Since p is prime and the factorization was nontrivial, Prolific writer 2 2 2 2 the factors 푎 + 푏 and 푥 + 푦 are equal to p. Proved four square Thus p can be written as a sum of two integer theorem in 1770 squares. Meticulous and shy Joseph-Louis Lagrange Born to a Jewish family German mathematician Number theorist Investigated number Mostly contributed to theory, Bessel functions, number theory and PDE’s algebras Work on quadratic Sickly mechanics influenced Einstein Also very sickly Adolf Hurwitz Rudolf Lipschitz 5 7/8/2011 Any natural number N can Hurwitz quaternions can be represented as be useful in a variety of 2 2 2 2 푁 = 푎 + 푏 + 푐 + 푑 ways! where a, b, c, and d are integers 6 .