Magic 15 Magic 15 Background Two Square Theorem Galois Finite Fields Magic 15: A STORY OVER 380 YEARS Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
From Shang Gao Magic 15 If a 2 + b 2 = c 2 , then ( a , b , c ) is called to be a Background Pythagorean triple . (3 , 4 , 5) , (5 , 12 , 13) are Pythagorean Two Square Theorem triples. If Pythagorean triple ( a , b , c ) satisfies Galois Finite gcd ( a , b , c ) = 1, then it is primitive . Fields Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
From Shang Gao Magic 15 If a 2 + b 2 = c 2 , then ( a , b , c ) is called to be a Background Pythagorean triple . (3 , 4 , 5) , (5 , 12 , 13) are Pythagorean Two Square Theorem triples. If Pythagorean triple ( a , b , c ) satisfies Galois Finite gcd ( a , b , c ) = 1, then it is primitive . Fields Proofs of (3 , 4 , 5) , (5 , 12 , 13) are primitive, but (6 , 8 , 10) is not. 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
From Shang Gao Magic 15 If a 2 + b 2 = c 2 , then ( a , b , c ) is called to be a Background Pythagorean triple . (3 , 4 , 5) , (5 , 12 , 13) are Pythagorean Two Square Theorem triples. If Pythagorean triple ( a , b , c ) satisfies Galois Finite gcd ( a , b , c ) = 1, then it is primitive . Fields Proofs of (3 , 4 , 5) , (5 , 12 , 13) are primitive, but (6 , 8 , 10) is not. 2-Square Theorem Question 1. How many primitive Pythagorean triples can Three Square you find? What is the largest one? Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
From Shang Gao Magic 15 If a 2 + b 2 = c 2 , then ( a , b , c ) is called to be a Background Pythagorean triple . (3 , 4 , 5) , (5 , 12 , 13) are Pythagorean Two Square Theorem triples. If Pythagorean triple ( a , b , c ) satisfies Galois Finite gcd ( a , b , c ) = 1, then it is primitive . Fields Proofs of (3 , 4 , 5) , (5 , 12 , 13) are primitive, but (6 , 8 , 10) is not. 2-Square Theorem Question 1. How many primitive Pythagorean triples can Three Square you find? What is the largest one? Theorem Four Square Answer: Infinitely many (so no ”Largest” can exist), since Theorem there are infinitely many rational points on x 2 + y 2 = 1 Universal Polynomials (via parametrization using the line x t + y = 1). Magic 15 Proof of Fifteen Theorem 15 and 290
What about x 2 + y 2 , ( x , y ) ∈ Z 2 ? Magic 15 Background Two Square Theorem Two Square Theorem. (1640, Fermat-1747, Euler) If Galois Finite p is a prime, then p = x 2 + y 2 ⇐ Fields ⇒ p ≡ 1( mod 4). Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
What about x 2 + y 2 , ( x , y ) ∈ Z 2 ? Magic 15 Background Two Square Theorem Two Square Theorem. (1640, Fermat-1747, Euler) If Galois Finite p is a prime, then p = x 2 + y 2 ⇐ Fields ⇒ p ≡ 1( mod 4). Proofs of Corollary. n = x 2 + y 2 ⇐ ⇒ n has no prime divisor of 2-Square Theorem the form 4 k + 3. Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
What about x 2 + y 2 , ( x , y ) ∈ Z 2 ? Magic 15 Background Two Square Theorem Two Square Theorem. (1640, Fermat-1747, Euler) If Galois Finite p is a prime, then p = x 2 + y 2 ⇐ Fields ⇒ p ≡ 1( mod 4). Proofs of Corollary. n = x 2 + y 2 ⇐ ⇒ n has no prime divisor of 2-Square Theorem the form 4 k + 3. Three Square Theorem Euler’s proof needs 5 lemmas, totally about 5 pages. Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Dedekind’s Proof of Two Square Theorem Magic 15 Background In 1894, Dedekind gave a proof of Two Square Theorem Two Square Theorem using Gauss integer ring Galois Finite Fields Z [ i ] = { a + bi | a , b ∈ Z } Proofs of 2-Square Theorem where i = √− 1 and its quotient Z [ i ] / ( p ) with p a prime Three Square Theorem number. Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Dedekind’s Proof of Two Square Theorem Magic 15 Background In 1894, Dedekind gave a proof of Two Square Theorem Two Square Theorem using Gauss integer ring Galois Finite Fields Z [ i ] = { a + bi | a , b ∈ Z } Proofs of 2-Square Theorem where i = √− 1 and its quotient Z [ i ] / ( p ) with p a prime Three Square Theorem number. Four Square Theorem What’s Z [ i ] / ( p )? Or easier, what’s Z / ( p )? Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Galois Finite Fields Magic 15 Z / ( p ) = { 0 , 1 , 2 , · · · , p − 1 } with natural ”+” and ” × ” is a finite field, called Galois finite field and denoted by Background GF ( p ) (or F p ), where the capital letter ”G” is E. Two Square Theorem Galois(1811-1831). Galois Finite Fields Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Galois Finite Fields Magic 15 Z / ( p ) = { 0 , 1 , 2 , · · · , p − 1 } with natural ”+” and ” × ” is a finite field, called Galois finite field and denoted by Background GF ( p ) (or F p ), where the capital letter ”G” is E. Two Square Theorem Galois(1811-1831). Galois Finite When p = 2, Z / (2) = { 0 , 1 } has the following ”+” and Fields ” × ” table: Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Galois Finite Fields Magic 15 Z / ( p ) = { 0 , 1 , 2 , · · · , p − 1 } with natural ”+” and ” × ” is a finite field, called Galois finite field and denoted by Background GF ( p ) (or F p ), where the capital letter ”G” is E. Two Square Theorem Galois(1811-1831). Galois Finite When p = 2, Z / (2) = { 0 , 1 } has the following ”+” and Fields ” × ” table: Proofs of 2-Square Theorem + 0 1 × 0 1 Three Square 0 0 1 0 0 0 Theorem 1 1 0 1 0 1 Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Galois Finite Fields Magic 15 Z / ( p ) = { 0 , 1 , 2 , · · · , p − 1 } with natural ”+” and ” × ” is a finite field, called Galois finite field and denoted by Background GF ( p ) (or F p ), where the capital letter ”G” is E. Two Square Theorem Galois(1811-1831). Galois Finite When p = 2, Z / (2) = { 0 , 1 } has the following ”+” and Fields ” × ” table: Proofs of 2-Square Theorem + 0 1 × 0 1 Three Square 0 0 1 0 0 0 Theorem 1 1 0 1 0 1 Four Square Theorem The above tables tell us exactly what parity is! Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Galois Finite Fields Magic 15 Z / ( p ) = { 0 , 1 , 2 , · · · , p − 1 } with natural ”+” and ” × ” is a finite field, called Galois finite field and denoted by Background GF ( p ) (or F p ), where the capital letter ”G” is E. Two Square Theorem Galois(1811-1831). Galois Finite When p = 2, Z / (2) = { 0 , 1 } has the following ”+” and Fields ” × ” table: Proofs of 2-Square Theorem + 0 1 × 0 1 Three Square 0 0 1 0 0 0 Theorem 1 1 0 1 0 1 Four Square Theorem The above tables tell us exactly what parity is! Universal Polynomials Can you partition all rational numbers into odd ones and Magic 15 even ones? Real or complex numbers?Can you partition all Proof of integral polynomials into odd ones and even ones? Fifteen Theorem 15 and 290
Elementary Geometry over F 2 Magic 15 What are the straight lines on the plane F 2 2 ? Background Two Square Theorem Galois Finite Fields Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Elementary Geometry over F 2 Magic 15 What are the straight lines on the plane F 2 2 ? Answer: Easy. Totally 6 (2 points = 1 line): Background x = 0 , x = 1 , y = 0 , y = 1 , x + y = 0 , x + y = 1. Two Square Theorem Galois Finite Fields Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Elementary Geometry over F 2 Magic 15 What are the straight lines on the plane F 2 2 ? Answer: Easy. Totally 6 (2 points = 1 line): Background x = 0 , x = 1 , y = 0 , y = 1 , x + y = 0 , x + y = 1. Two Square Theorem What are the circles on the plane F 2 ? How many are they? Galois Finite Fields Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
Elementary Geometry over F 2 Magic 15 What are the straight lines on the plane F 2 2 ? Answer: Easy. Totally 6 (2 points = 1 line): Background x = 0 , x = 1 , y = 0 , y = 1 , x + y = 0 , x + y = 1. Two Square Theorem What are the circles on the plane F 2 ? How many are they? Galois Finite Answer: A little surprising. Only 2. ( Comparing: ”More” Fields circles than lines in R 2 .) Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15 and 290
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