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%3 N_20200621104155a_ColinWright    By ColinWright 2020/06/21 @ 10:41:55a -------------------------------- Thinking about Group Theory ...        (select only this node)     N_20200621110158a_ColinWright    By ColinWright 2020/06/21 @ 11:01:58a -------------------------------- In the integers, x^2+1=0 has no solutions        (select only this node)     N_20200621104155a_ColinWright->N_20200621110158a_ColinWright N_20200621140925a_ColinWright    By ColinWright 2020/06/21 @ 14:09:25a -------------------------------- Groups        (select only this node)     N_20200621104155a_ColinWright->N_20200621140925a_ColinWright N_20200621105050c_ColinWright    By ColinWright 2020/06/21 @ 10:50:50c -------------------------------- Wilson's Theorem        (select only this node)     N_20200621104155a_ColinWright->N_20200621105050c_ColinWright N_20200621104312a_ColinWright    By ColinWright 2020/06/21 @ 10:43:12a -------------------------------- We can show that if n=3 (mod 4) then n is not the sum of two squares        (select only this node)     N_20200621104436a_ColinWright    By ColinWright 2020/06/21 @ 10:44:36a -------------------------------- Proof: Working mod 4, squares are either 0 or 1, so adding two of them can never give 3 (mod 4)        (select only this node)     N_20200621104312a_ColinWright->N_20200621104436a_ColinWright N_20200621104436b_ColinWright    By ColinWright 2020/06/21 @ 10:44:36b -------------------------------- A similar approach lets us show that 100000007 (10^7+7) is not the sum of three squares (working mod 8)        (select only this node)     N_20200621104436a_ColinWright->N_20200621104436b_ColinWright N_20200621104600a_ColinWright    By ColinWright 2020/06/21 @ 10:46:00a -------------------------------- Claim: If p=3 (mod 4) is prime, then there is no solution to u^2=-1 (mod p)        (select only this node)     N_20200621104744a_ColinWright    By ColinWright 2020/06/21 @ 10:47:44a -------------------------------- Proof: Suppose we have u^2=-1. Then {1,u,u^2,u^3} is a sub-group of Z_p^x. So 4 divides the size of Z_p^x, which is p-1. Hence p=1 (mod 4).        (select only this node)     N_20200621104600a_ColinWright->N_20200621104744a_ColinWright N_20200621104600a_ColinWright->N_20200621140925a_ColinWright N_20200621132917a_ColinWright    By ColinWright 2020/06/21 @ 13:29:17a -------------------------------- TARGET        (select only this node)     N_20200621104744a_ColinWright->N_20200621132917a_ColinWright N_20200621104852a_ColinWright    By ColinWright 2020/06/21 @ 10:48:52a -------------------------------- A group generated by one element is cyclic.        (select only this node)     N_20200621104852b_ColinWright    By ColinWright 2020/06/21 @ 10:48:52b -------------------------------- Lagranges Theorem        (select only this node)     N_20200621104852b_ColinWright->N_20200621104744a_ColinWright N_20200621110607a_ColinWright    By ColinWright 2020/06/21 @ 11:06:07a -------------------------------- Proof: Given H a subgroup of G (finite)        (select only this node)     N_20200621104852b_ColinWright->N_20200621110607a_ColinWright N_20200621105050a_ColinWright    By ColinWright 2020/06/21 @ 10:50:50a -------------------------------- Claim: If p=1 (mod 4) then u^2=-1 (mod p) has a solution        (select only this node)     N_20200621105050b_ColinWright    By ColinWright 2020/06/21 @ 10:50:50b -------------------------------- Proof: ((p-1)/2)^2=-1 (mod 4) via Wilson's Theorem        (select only this node)     N_20200621105050a_ColinWright->N_20200621105050b_ColinWright N_20200621105310b_ColinWright    By ColinWright 2020/06/21 @ 10:53:10b -------------------------------- Proof: Let u^2=-1 and consider all a+bu with 0<=a,b<sqrt(p). There must be a duplicate, and we chase the arithmetic.        (select only this node)     N_20200621105050a_ColinWright->N_20200621105310b_ColinWright N_20200621150943a_ColinWright    By ColinWright 2020/06/21 @ 15:09:43a -------------------------------- For integers n>1, (n-1)!+1 is a multiple of n if and only if n is prime.        (select only this node)     N_20200621105050c_ColinWright->N_20200621150943a_ColinWright N_20200621105310a_ColinWright    By ColinWright 2020/06/21 @ 10:53:10a -------------------------------- So it seems that if p=1 (mod 4) is prime, then p=a^2+b^2. The question: is this always true?        (select only this node)     N_20200621105310a_ColinWright->N_20200621105310b_ColinWright N_20200621165715a_ColinWright    By ColinWright 2020/06/21 @ 16:57:15a -------------------------------- Proof by magic ...        (select only this node)     N_20200621105310a_ColinWright->N_20200621165715a_ColinWright N_20200621110158d_ColinWright    By ColinWright 2020/06/21 @ 11:01:58d -------------------------------- We can explore the values of x^2+1 and see if there's anything interesting        (select only this node)     N_20200621110158a_ColinWright->N_20200621110158d_ColinWright N_20200621120414a_ColinWright    By ColinWright 2020/06/21 @ 12:04:14a -------------------------------- Square root of -1        (select only this node)     N_20200621110158a_ColinWright->N_20200621120414a_ColinWright N_20200621110158b_ColinWright    By ColinWright 2020/06/21 @ 11:01:58b -------------------------------- We can simply accept it        (select only this node)     N_20200621110158c_ColinWright    By ColinWright 2020/06/21 @ 11:01:58c -------------------------------- We can invent the Complex Numbers        (select only this node)     N_20200621110158e_ColinWright    By ColinWright 2020/06/21 @ 11:01:58e -------------------------------- x=0 : x^2+1 = 1 x=1 : x^2+1 = 2 -> prime x=2 : x^2+1 = 5 -> prime x=3 : x^2+1 = 10 -> prime * 2 x=4 : x^2+1 = 17 -> prime x=5 : x^2+1 = 26 -> prime * 2 x=6 : x^2+1 = 37 -> prime x=7 : x^2+1 = 50 x=8 : x^2+1 = 65 -> 5 x 13 x=9 : x^2+1 = 82 -> prime * 2 x=10 : x^2+1 = 101 -> prime x=11 : x^2+1 = 122 -> prime * 2 x=12 : x^2+1 = 145 -> 5 x 29 x=13 : x^2+1 = 170 -> 2 x 5 x 17 x=14 : x^2+1 = 197 -> prime x=15 : x^2+1 = 226 -> prime * 2 x=16 : x^2+1 = 257 -> prime        (select only this node)     N_20200621110158d_ColinWright->N_20200621110158e_ColinWright N_20200621133734a_ColinWright    By ColinWright 2020/06/21 @ 13:37:34a -------------------------------- C4 is a sub-group of S4        (select only this node)     N_20200621142419a_ColinWright    By ColinWright 2020/06/21 @ 14:24:19a -------------------------------- Consider the size of a sub-group compared with the super-group ...        (select only this node)     N_20200621133734a_ColinWright->N_20200621142419a_ColinWright N_20200621133734c_ColinWright    By ColinWright 2020/06/21 @ 13:37:34c -------------------------------- S4 is isomorphic to the rotations of a cube        (select only this node)     N_20200621134033a_ColinWright    By ColinWright 2020/06/21 @ 13:40:33a -------------------------------- The groups of permutations on 4 objects is S4        (select only this node)     N_20200621140948a_ColinWright    By ColinWright 2020/06/21 @ 14:09:48a -------------------------------- What is a group?        (select only this node)     N_20200621134033a_ColinWright->N_20200621140948a_ColinWright N_20200621145006a_ColinWright    By ColinWright 2020/06/21 @ 14:50:06a -------------------------------- These both have 24 elements        (select only this node)     N_20200621134033a_ColinWright->N_20200621145006a_ColinWright N_20200621133734e_ColinWright    By ColinWright 2020/06/21 @ 13:37:34e -------------------------------- Permutation Diagrams        (select only this node)     N_20200621134033a_ColinWright->N_20200621133734e_ColinWright N_20200621124907a_ColinWright    By ColinWright 2020/06/21 @ 12:49:07a -------------------------------- These we have already: 2, 5, 17, 37, 101, ...        (select only this node)     N_20200621110158e_ColinWright->N_20200621124907a_ColinWright N_20200621121212a_ColinWright    By ColinWright 2020/06/21 @ 12:12:12a -------------------------------- Pick one of the primes ... this shows that modulo that prime, there is a square root of -1.        (select only this node)     N_20200621110158e_ColinWright->N_20200621121212a_ColinWright N_20200625140205a_ColinWright    By ColinWright 2020/06/25 @ 14:02:05a -------------------------------- A coset of H in G is the set we get by multiplying every element of H by some element g in G.        (select only this node)     N_20200621110607a_ColinWright->N_20200625140205a_ColinWright N_20200621110607b_ColinWright    By ColinWright 2020/06/21 @ 11:06:07b -------------------------------- The cosets of H are disjoint        (select only this node)     N_20200621110607c_ColinWright    By ColinWright 2020/06/21 @ 11:06:07c -------------------------------- The cosets of H are all the same size        (select only this node)     N_20200621110607b_ColinWright->N_20200621110607c_ColinWright N_20200621110607e_ColinWright    By ColinWright 2020/06/21 @ 11:06:07e -------------------------------- Every element of G is in exactly one coset of H        (select only this node)     N_20200621110607b_ColinWright->N_20200621110607e_ColinWright N_20200621110607f_ColinWright    By ColinWright 2020/06/21 @ 11:06:07f -------------------------------- The cosets of H "tile" G        (select only this node)     N_20200621110607c_ColinWright->N_20200621110607f_ColinWright N_20200621110607d_ColinWright    By ColinWright 2020/06/21 @ 11:06:07d -------------------------------- Every element of G is in a coset of H        (select only this node)     N_20200621110607d_ColinWright->N_20200621110607e_ColinWright N_20200625140702a_ColinWright    By ColinWright 2020/06/25 @ 14:07:02a -------------------------------- Let g be in G, then g is in the coset gH        (select only this node)     N_20200621110607d_ColinWright->N_20200625140702a_ColinWright N_20200621110607e_ColinWright->N_20200621110607f_ColinWright N_20200621110607g_ColinWright    By ColinWright 2020/06/21 @ 11:06:07g -------------------------------- QED (Lagrange's Theorem)        (select only this node)     N_20200621110607f_ColinWright->N_20200621110607g_ColinWright N_20200621134033b_ColinWright    By ColinWright 2020/06/21 @ 13:40:33b -------------------------------- The symmetries of a mattress is V4        (select only this node)     N_20200621134033b_ColinWright->N_20200621134033a_ColinWright N_20200621112247a_ColinWright    By ColinWright 2020/06/21 @ 11:22:47a -------------------------------- Composite case: We prove that if n is composite then n does not divide (n-1)!+1        (select only this node)     N_20200621112247b_ColinWright    By ColinWright 2020/06/21 @ 11:22:47b -------------------------------- If n is composite then we have three possibilities:        (select only this node)     N_20200621112247a_ColinWright->N_20200621112247b_ColinWright N_20200621112247e_ColinWright    By ColinWright 2020/06/21 @ 11:22:47e -------------------------------- If n=4, n does not divide (n-1)!+1.        (select only this node)     N_20200621112247b_ColinWright->N_20200621112247e_ColinWright N_20200621112247c_ColinWright    By ColinWright 2020/06/21 @ 11:22:47c -------------------------------- If n=a.b with a<b, then a and b both turn up in (n-1)!, so (n-1)! is a multiple of n, so n does not divide (n-1)!+1.        (select only this node)     N_20200621112247b_ColinWright->N_20200621112247c_ColinWright N_20200621112247d_ColinWright    By ColinWright 2020/06/21 @ 11:22:47d -------------------------------- If n=p^2 with p>2, then p and 2p both turn up in (n-1)!, so (n-1)! is a multiple of n, so n does not divide (n-1)!+1.        (select only this node)     N_20200621112247b_ColinWright->N_20200621112247d_ColinWright N_20200621112247f_ColinWright    By ColinWright 2020/06/21 @ 11:22:47f -------------------------------- So if n is composite, n does not divide (n-1)!+1.        (select only this node)     N_20200621112247c_ColinWright->N_20200621112247f_ColinWright N_20200621112247d_ColinWright->N_20200621112247f_ColinWright N_20200621112247e_ColinWright->N_20200621112247f_ColinWright N_20200621112857a_ColinWright    By ColinWright 2020/06/21 @ 11:28:57a -------------------------------- QED (Wilson's Theorem)        (select only this node)     N_20200621112247f_ColinWright->N_20200621112857a_ColinWright N_20200621112808a_ColinWright    By ColinWright 2020/06/21 @ 11:28:08a -------------------------------- Prime case: We prove that if p is prime, then p *does* divide (p-1)!+1.        (select only this node)     N_20200621112808b_ColinWright    By ColinWright 2020/06/21 @ 11:28:08b -------------------------------- Clearly true for p=2 and p=3, so we start with p=5.        (select only this node)     N_20200621112808a_ColinWright->N_20200621112808b_ColinWright N_20200621112808c_ColinWright    By ColinWright 2020/06/21 @ 11:28:08c -------------------------------- Consider 2, 3, ... (p-2). Each of these has another as an inverse.        (select only this node)     N_20200621112808b_ColinWright->N_20200621112808c_ColinWright N_20200621112808d_ColinWright    By ColinWright 2020/06/21 @ 11:28:08d -------------------------------- Suppose x^2=(k.p)+1. Then (x-1)(x+1)=kp, so p divides either x-1 or x+1.        (select only this node)     N_20200621112808c_ColinWright->N_20200621112808d_ColinWright N_20200621150353a_ColinWright    By ColinWright 2020/06/21 @ 15:03:53a -------------------------------- This is *not* obvious, but it means that we have an equivalent to division when working modulo a prime.        (select only this node)     N_20200621112808c_ColinWright->N_20200621150353a_ColinWright N_20200621112808e_ColinWright    By ColinWright 2020/06/21 @ 11:28:08e -------------------------------- Pairing each with its inverse we see that they all cancel out.        (select only this node)     N_20200621112808c_ColinWright->N_20200621112808e_ColinWright N_20200621134033c_ColinWright    By ColinWright 2020/06/21 @ 13:40:33c -------------------------------- The cyclic group on four elements is C4        (select only this node)     N_20200621112808f_ColinWright    By ColinWright 2020/06/21 @ 11:28:08f -------------------------------- So in (p-1)! everything cancels except (p-1).        (select only this node)     N_20200621112808e_ColinWright->N_20200621112808f_ColinWright N_20200621112808g_ColinWright    By ColinWright 2020/06/21 @ 11:28:08g -------------------------------- Hence (p-1)!=-1 (mod p)        (select only this node)     N_20200621112808f_ColinWright->N_20200621112808g_ColinWright N_20200621112808g_ColinWright->N_20200621112857a_ColinWright N_20200621112857a_ColinWright->N_20200621105050b_ColinWright N_20200621133734d_ColinWright    By ColinWright 2020/06/21 @ 13:37:34d -------------------------------- Cayley Tables        (select only this node)     N_20200621134033d_ColinWright    By ColinWright 2020/06/21 @ 13:40:33d -------------------------------- Rotations of a square is C4        (select only this node)     N_20200621143216a_ColinWright    By ColinWright 2020/06/21 @ 14:32:16a -------------------------------- Rotations of an n-gon is Cn        (select only this node)     N_20200621134033d_ColinWright->N_20200621143216a_ColinWright N_20200621134033d_ColinWright->N_20200621134033c_ColinWright N_20200621134033d_ColinWright->N_20200621134033a_ColinWright N_20200621110607g_ColinWright->N_20200621104744a_ColinWright N_20200621114035a_ColinWright    By ColinWright 2020/06/21 @ 11:40:35a -------------------------------- Try to do n=17 by hand.        (select only this node)     N_20200621114035b_ColinWright    By ColinWright 2020/06/21 @ 11:40:35b -------------------------------- Observe that the 2 and 9 "cancel out".        (select only this node)     N_20200621114035a_ColinWright->N_20200621114035b_ColinWright N_20200621114035d_ColinWright    By ColinWright 2020/06/21 @ 11:40:35d -------------------------------- Ditto (8,15), (10,12), (11,14) ...        (select only this node)     N_20200621114035b_ColinWright->N_20200621114035d_ColinWright N_20200621114035c_ColinWright    By ColinWright 2020/06/21 @ 11:40:35c -------------------------------- Ditto (3,6), (4,13), (5,7) ...        (select only this node)     N_20200621114035b_ColinWright->N_20200621114035c_ColinWright N_20200621114035e_ColinWright    By ColinWright 2020/06/21 @ 11:40:35e -------------------------------- So everything cancels except the 16, which is -1. (mod 17)        (select only this node)     N_20200621114035c_ColinWright->N_20200621114035e_ColinWright N_20200621114035d_ColinWright->N_20200621114035e_ColinWright N_20200621151349a_ColinWright    By ColinWright 2020/06/21 @ 15:13:49a -------------------------------- So let's prove it.        (select only this node)     N_20200621114035e_ColinWright->N_20200621151349a_ColinWright N_20200621114637a_ColinWright    By ColinWright 2020/06/21 @ 11:46:37a -------------------------------- This shows that considering just the remainders after division by something (in this case a prime) can be really powerful.        (select only this node)     N_20200621114035e_ColinWright->N_20200621114637a_ColinWright N_20200621114637c_ColinWright    By ColinWright 2020/06/21 @ 11:46:37c -------------------------------- So we have the concept of "Modulo Arithmetic"        (select only this node)     N_20200621114637a_ColinWright->N_20200621114637c_ColinWright N_20200621114637b_ColinWright    By ColinWright 2020/06/21 @ 11:46:37b -------------------------------- (This is the same as the process of "casting out 9s")        (select only this node)     N_20200621114637a_ColinWright->N_20200621114637b_ColinWright N_20200621114637d_ColinWright    By ColinWright 2020/06/21 @ 11:46:37d -------------------------------- We have some number - the modulus - and any two numbers that differ by a multiple of that are considered the same.        (select only this node)     N_20200621114637c_ColinWright->N_20200621114637d_ColinWright N_20200621114637e_ColinWright    By ColinWright 2020/06/21 @ 11:46:37e -------------------------------- Think of putting the integers on a big circle of some size ... number that fall in the same place are thought of as "the same thing".        (select only this node)     N_20200621114637d_ColinWright->N_20200621114637e_ColinWright N_20200621142755a_ColinWright    By ColinWright 2020/06/21 @ 14:27:55a -------------------------------- Addition modulo N is a cyclic group of size n.        (select only this node)     N_20200621114637e_ColinWright->N_20200621142755a_ColinWright N_20200621114637e_ColinWright->N_20200621121212a_ColinWright N_20200621114637e_ColinWright->N_20200621105050a_ColinWright N_20200621114637e_ColinWright->N_20200621105310a_ColinWright N_20200621114637e_ColinWright->N_20200621104312a_ColinWright N_20200621120229a_ColinWright    By ColinWright 2020/06/21 @ 12:02:29a -------------------------------- And again checking by hand, we can work out that these are *also* the sum of two squares.        (select only this node)     N_20200621120229a_ColinWright->N_20200621105310a_ColinWright N_20200621120414a_ColinWright->N_20200621110158c_ColinWright N_20200621120414a_ColinWright->N_20200621110158b_ColinWright N_20200621121235a_ColinWright    By ColinWright 2020/06/21 @ 12:12:35a -------------------------------- What about the other primes?        (select only this node)     N_20200621121212a_ColinWright->N_20200621121235a_ColinWright N_20200621121235a_ColinWright->N_20200621124907a_ColinWright N_20200621124907b_ColinWright    By ColinWright 2020/06/21 @ 12:49:07b -------------------------------- Checking by hand, we can also see that these have a square root of -1: 13, 29, 41, 53, 61, ...        (select only this node)     N_20200621121235a_ColinWright->N_20200621124907b_ColinWright N_20200621124907c_ColinWright    By ColinWright 2020/06/21 @ 12:49:07c -------------------------------- Checking by hand we can confirm that these do not have a square root of -1: 3, 7, 11, 19, 23, 31, 43, 47, 59, ...        (select only this node)     N_20200621121235a_ColinWright->N_20200621124907c_ColinWright N_20200621150353b_ColinWright    By ColinWright 2020/06/21 @ 15:03:53b -------------------------------- Since every element a has an inverse b, we can solve equations of the form a.x = c.        (select only this node)     N_20200621150353a_ColinWright->N_20200621150353b_ColinWright N_20200621122115a_ColinWright    By ColinWright 2020/06/21 @ 12:21:15a -------------------------------- Since 0<x<p, that means either x=1 or x=p-1, both of which we have excluded.        (select only this node)     N_20200621112808d_ColinWright->N_20200621122115a_ColinWright N_20200621125330a_ColinWright    By ColinWright 2020/06/21 @ 12:53:30a -------------------------------- These primes are all 1 more than a multiple of 4.        (select only this node)     N_20200621124907a_ColinWright->N_20200621125330a_ColinWright N_20200621124907a_ColinWright->N_20200621124907b_ColinWright N_20200621161845a_ColinWright    By ColinWright 2020/06/21 @ 16:18:45a -------------------------------- So they are x^2+1, but let's generalise a little:        (select only this node)     N_20200621124907a_ColinWright->N_20200621161845a_ColinWright N_20200621124907b_ColinWright->N_20200621125330a_ColinWright N_20200621124907b_ColinWright->N_20200621120229a_ColinWright N_20200621124907c_ColinWright->N_20200621104312a_ColinWright N_20200621125330b_ColinWright    By ColinWright 2020/06/21 @ 12:53:30b -------------------------------- Does every prime of the form 4k+1 have a square root of -1?        (select only this node)     N_20200621125330a_ColinWright->N_20200621125330b_ColinWright N_20200621125330b_ColinWright->N_20200621104600a_ColinWright N_20200621125330b_ColinWright->N_20200621105050a_ColinWright N_20200621133734b_ColinWright    By ColinWright 2020/06/21 @ 13:37:34b -------------------------------- V4 is a sub-group of S4        (select only this node)     N_20200621133734b_ColinWright->N_20200621142419a_ColinWright N_20200621140925a_ColinWright->N_20200621134033d_ColinWright N_20200621140925a_ColinWright->N_20200621134033b_ColinWright N_20200621143650a_ColinWright    By ColinWright 2020/06/21 @ 14:36:50a -------------------------------- Rotations of a cube        (select only this node)     N_20200621140925a_ColinWright->N_20200621143650a_ColinWright N_20200621140948b_ColinWright    By ColinWright 2020/06/21 @ 14:09:48b -------------------------------- What is a subgroup?        (select only this node)     N_20200621140948a_ColinWright->N_20200621140948b_ColinWright N_20200621140948a_ColinWright->N_20200621133734d_ColinWright N_20200621140948b_ColinWright->N_20200621133734b_ColinWright N_20200621140948b_ColinWright->N_20200621133734a_ColinWright N_20200621140948b_ColinWright->N_20200621104852b_ColinWright N_20200621142419a_ColinWright->N_20200621104852b_ColinWright N_20200621142755a_ColinWright->N_20200621104852a_ColinWright N_20200621143216a_ColinWright->N_20200621142755a_ColinWright N_20200621143650a_ColinWright->N_20200621145006a_ColinWright N_20200621150353c_ColinWright    By ColinWright 2020/06/21 @ 15:03:53c -------------------------------- Usually we would divide both sides by by a, but here instead we (pre-)multiply both sides by b.        (select only this node)     N_20200621150353b_ColinWright->N_20200621150353c_ColinWright N_20200621145148a_ColinWright    By ColinWright 2020/06/21 @ 14:51:48a -------------------------------- Are they the same?        (select only this node)     N_20200621145006a_ColinWright->N_20200621145148a_ColinWright N_20200621145148b_ColinWright    By ColinWright 2020/06/21 @ 14:51:48b -------------------------------- Yes        (select only this node)     N_20200621145148a_ColinWright->N_20200621145148b_ColinWright N_20200621145148b_ColinWright->N_20200621133734c_ColinWright N_20200621150353d_ColinWright    By ColinWright 2020/06/21 @ 15:03:53d -------------------------------- Then we have b.(a.x)=b.c, and by associativity we have (b.a).x=b.c, so x=b.c        (select only this node)     N_20200621150353c_ColinWright->N_20200621150353d_ColinWright N_20200621150353e_ColinWright    By ColinWright 2020/06/21 @ 15:03:53e -------------------------------- Multiplying by b is giving us the same power as dividing by a. They are multiplicative inverses.        (select only this node)     N_20200621150353d_ColinWright->N_20200621150353e_ColinWright N_20200621150943b_ColinWright    By ColinWright 2020/06/21 @ 15:09:43b -------------------------------- Check a few by hand:        (select only this node)     N_20200621150943e_ColinWright    By ColinWright 2020/06/21 @ 15:09:43e -------------------------------- Composites:        (select only this node)     N_20200621150943b_ColinWright->N_20200621150943e_ColinWright N_20200621150943c_ColinWright    By ColinWright 2020/06/21 @ 15:09:43c -------------------------------- Primes:        (select only this node)     N_20200621150943b_ColinWright->N_20200621150943c_ColinWright N_20200621150943d_ColinWright    By ColinWright 2020/06/21 @ 15:09:43d -------------------------------- n=2 -> 2, multiple n=3 -> 3, multiple n=5 -> 25, multiple n=7 -> 721, multiple        (select only this node)     N_20200621150943c_ColinWright->N_20200621150943d_ColinWright N_20200621150943f_ColinWright    By ColinWright 2020/06/21 @ 15:09:43f -------------------------------- n=4 -> 7, not multiple n=6 -> 121, not multiple n=8 -> 5041, not multiple n=9 -> 40321, not multiple        (select only this node)     N_20200621150943e_ColinWright->N_20200621150943f_ColinWright N_20200621150943g_ColinWright    By ColinWright 2020/06/21 @ 15:09:43g -------------------------------- So far, so good.        (select only this node)     N_20200621150943g_ColinWright->N_20200621114035a_ColinWright N_20200621151349a_ColinWright->N_20200621112808a_ColinWright N_20200621151349a_ColinWright->N_20200621112247a_ColinWright N_20200621150943a_ColinWright->N_20200621151349a_ColinWright N_20200621150943a_ColinWright->N_20200621150943b_ColinWright N_20200621150943d_ColinWright->N_20200621150943g_ColinWright N_20200621150943f_ColinWright->N_20200621112247a_ColinWright N_20200621150943f_ColinWright->N_20200621150943g_ColinWright N_20200621161845b_ColinWright    By ColinWright 2020/06/21 @ 16:18:45b -------------------------------- They are the sum of two squares.        (select only this node)     N_20200621161845a_ColinWright->N_20200621161845b_ColinWright N_20200621161845c_ColinWright    By ColinWright 2020/06/21 @ 16:18:45c -------------------------------- (Where one of the squares is 1)        (select only this node)     N_20200621161845b_ColinWright->N_20200621161845c_ColinWright N_20200621161845b_ColinWright->N_20200621120229a_ColinWright N_20200621165715b_ColinWright    By ColinWright 2020/06/21 @ 16:57:15b -------------------------------- Guess values of b until b^((p-1)/2)=-1(mod p).        (select only this node)     N_20200621165715a_ColinWright->N_20200621165715b_ColinWright N_20200621165715c_ColinWright    By ColinWright 2020/06/21 @ 16:57:15c -------------------------------- Hence z=b^((p-1)/4) is a square-root of -1.        (select only this node)     N_20200621165715b_ColinWright->N_20200621165715c_ColinWright N_20200621165715e_ColinWright    By ColinWright 2020/06/21 @ 16:57:15e -------------------------------- (z+i)(z-i)=z^2+1=0(mod p).        (select only this node)     N_20200621165715c_ColinWright->N_20200621165715e_ColinWright N_20200621165715d_ColinWright    By ColinWright 2020/06/21 @ 16:57:15d -------------------------------- (This shows that there is a u with u^2+1=0 (mod p))        (select only this node)     N_20200621165715c_ColinWright->N_20200621165715d_ColinWright N_20200621165715f_ColinWright    By ColinWright 2020/06/21 @ 16:57:15f -------------------------------- Compute GCD(z+i,p) in Z[i], giving a+b.i        (select only this node)     N_20200621165715e_ColinWright->N_20200621165715f_ColinWright N_20200621165715g_ColinWright    By ColinWright 2020/06/21 @ 16:57:15g -------------------------------- Then a^2+b^2=p.        (select only this node)     N_20200621165715f_ColinWright->N_20200621165715g_ColinWright N_20200625140205a_ColinWright->N_20200621110607b_ColinWright N_20200625140205a_ColinWright->N_20200621110607c_ColinWright N_20200625140205b_ColinWright    By ColinWright 2020/06/25 @ 14:02:05b -------------------------------- If g is in H then the coset is simply H        (select only this node)     N_20200625140205a_ColinWright->N_20200625140205b_ColinWright N_20200625140205a_ColinWright->N_20200621110607d_ColinWright N_20200625140205c_ColinWright    By ColinWright 2020/06/25 @ 14:02:05c -------------------------------- If g is *not* in H then what we get is not a group, because it does not contain the identity element.        (select only this node)     N_20200625140205b_ColinWright->N_20200625140205c_ColinWright