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Discussion: PotpourriOfPythagorasAndPrimes
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%3 N_20210322115905a_ColinWright    By ColinWright 2021/03/22 @ 11:59:05a -------------------------------- Pythagoras, primes, squares, triangles, circles, area, and more        (select only this node)     N_20210322121527c_ColinWright    By ColinWright 2021/03/22 @ 12:15:27c -------------------------------- Squares        (select only this node)     N_20210322115905a_ColinWright->N_20210322121527c_ColinWright N_20210322121527a_ColinWright    By ColinWright 2021/03/22 @ 12:15:27a -------------------------------- Pythagoras        (select only this node)     N_20210322115905a_ColinWright->N_20210322121527a_ColinWright N_20210322121527b_ColinWright    By ColinWright 2021/03/22 @ 12:15:27b -------------------------------- Primes        (select only this node)     N_20210322115905a_ColinWright->N_20210322121527b_ColinWright N_20210322115905b_ColinWright    By ColinWright 2021/03/22 @ 11:59:05b -------------------------------- In a right-angled triangle, the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side        (select only this node)     N_20210322115905d_ColinWright    By ColinWright 2021/03/22 @ 11:59:05d -------------------------------- A^2+B^2=C^2, for a suitably labelled right-angled triangle.        (select only this node)     N_20210322115905b_ColinWright->N_20210322115905d_ColinWright N_20210322115905c_ColinWright    By ColinWright 2021/03/22 @ 11:59:05c -------------------------------- The total area of squares constructed on the two shorter sides is the same as the area of the square constructed on the longest side        (select only this node)     N_20210322115905c_ColinWright->N_20210322115905d_ColinWright N_20210324090708a_ColinWright    By ColinWright 2021/03/24 @ 09:07:08a -------------------------------- Sum of two squares ...        (select only this node)     N_20210322115905d_ColinWright->N_20210324090708a_ColinWright N_20210322120147a_ColinWright    By ColinWright 2021/03/22 @ 12:01:47a -------------------------------- So this is an "if and only if" theorem ... often one direction is quoted, but the other is used.        (select only this node)     N_20210322115905d_ColinWright->N_20210322120147a_ColinWright N_20210322120320a_ColinWright    By ColinWright 2021/03/22 @ 12:03:20a -------------------------------- Right-angled triangle ... pick a specific example ... I'll use 8:15:17        (select only this node)     N_20210322115905d_ColinWright->N_20210322120320a_ColinWright N_20210322120045a_ColinWright    By ColinWright 2021/03/22 @ 12:00:45a -------------------------------- If, in a triangle T with sides A, B, and C, we have that A^2+B^2=C^2, then T is a right-angled triangle.        (select only this node)     N_20210322120045a_ColinWright->N_20210324090708a_ColinWright N_20210322120045a_ColinWright->N_20210322120147a_ColinWright N_20210322120045a_ColinWright->N_20210322120320a_ColinWright N_20210322120147b_ColinWright    By ColinWright 2021/03/22 @ 12:01:47b -------------------------------- Yes, both are true, but it's worth noting.        (select only this node)     N_20210322120147a_ColinWright->N_20210322120147b_ColinWright N_20210322120320b_ColinWright    By ColinWright 2021/03/22 @ 12:03:20b -------------------------------- What is the radius of the circumcircle?        (select only this node)     N_20210322120320a_ColinWright->N_20210322120320b_ColinWright N_20210322120320c_ColinWright    By ColinWright 2021/03/22 @ 12:03:20c -------------------------------- What is the radius of the incircle?        (select only this node)     N_20210322120320b_ColinWright->N_20210322120320c_ColinWright N_20210322120358a_ColinWright    By ColinWright 2021/03/22 @ 12:03:58a -------------------------------- (Calculation left unspecified)        (select only this node)     N_20210322120320b_ColinWright->N_20210322120358a_ColinWright N_20210322121359a_ColinWright    By ColinWright 2021/03/22 @ 12:13:59a -------------------------------- Does this surprise you?        (select only this node)     N_20210322120320c_ColinWright->N_20210322121359a_ColinWright N_20210322121359b_ColinWright    By ColinWright 2021/03/22 @ 12:13:59b -------------------------------- What answers are possible for other Pythagorean triangles?        (select only this node)     N_20210322121359a_ColinWright->N_20210322121359b_ColinWright N_20210322121359c_ColinWright    By ColinWright 2021/03/22 @ 12:13:59c -------------------------------- Can you prove that?        (select only this node)     N_20210322121359b_ColinWright->N_20210322121359c_ColinWright N_20210322121359f_ColinWright    By ColinWright 2021/03/22 @ 12:13:59f -------------------------------- Convince an enemy.        (select only this node)     N_20210322121359c_ColinWright->N_20210322121359f_ColinWright N_20210322121359e_ColinWright    By ColinWright 2021/03/22 @ 12:13:59e -------------------------------- Convince a friend;        (select only this node)     N_20210322121359c_ColinWright->N_20210322121359e_ColinWright N_20210322121359d_ColinWright    By ColinWright 2021/03/22 @ 12:13:59d -------------------------------- Convince yourself;        (select only this node)     N_20210322121359c_ColinWright->N_20210322121359d_ColinWright N_20210322121359d_ColinWright->N_20210322121359e_ColinWright N_20210322121359e_ColinWright->N_20210322121359f_ColinWright N_20210322121527a_ColinWright->N_20210322115905c_ColinWright N_20210322121527a_ColinWright->N_20210322120045a_ColinWright N_20210322121527a_ColinWright->N_20210322115905b_ColinWright N_20210322124629a_ColinWright    By ColinWright 2021/03/22 @ 12:46:29a -------------------------------- Take some collection of tokens, and try to form them into a (non-trivial) rectangle.        (select only this node)     N_20210322121527b_ColinWright->N_20210322124629a_ColinWright N_20210324085639a_ColinWright    By ColinWright 2021/03/24 @ 08:56:39a -------------------------------- Square and Primes        (select only this node)     N_20210322121527c_ColinWright->N_20210324085639a_ColinWright N_20210322122555a_ColinWright    By ColinWright 2021/03/22 @ 12:25:55a -------------------------------- Write down the odd numbers starting from 1        (select only this node)     N_20210322121527c_ColinWright->N_20210322122555a_ColinWright N_20210322122555b_ColinWright    By ColinWright 2021/03/22 @ 12:25:55b -------------------------------- Compute the running total.        (select only this node)     N_20210322122555a_ColinWright->N_20210322122555b_ColinWright N_20210322122624a_ColinWright    By ColinWright 2021/03/22 @ 12:26:24a -------------------------------- 1, 3, 5, 7, 9, 11, ...        (select only this node)     N_20210322122555a_ColinWright->N_20210322122624a_ColinWright N_20210324090434b_ColinWright    By ColinWright 2021/03/24 @ 09:04:34b -------------------------------- Find another        (select only this node)     N_20210324090434c_ColinWright    By ColinWright 2021/03/24 @ 09:04:34c -------------------------------- Let's classify primes accordingly, Red or Green.        (select only this node)     N_20210324090434b_ColinWright->N_20210324090434c_ColinWright N_20210322122555d_ColinWright    By ColinWright 2021/03/22 @ 12:25:55d -------------------------------- So going from a square A^2 to the next square (A+1)^2 is always adding an odd number.        (select only this node)     N_20210322122555f_ColinWright    By ColinWright 2021/03/22 @ 12:25:55f -------------------------------- What if we deliberately choose a case where that odd number is itself a square?        (select only this node)     N_20210322122555d_ColinWright->N_20210322122555f_ColinWright N_20210322122555e_ColinWright    By ColinWright 2021/03/22 @ 12:25:55e -------------------------------- (This is obvious to many, but not all.)        (select only this node)     N_20210322122555d_ColinWright->N_20210322122555e_ColinWright N_20210322122555g_ColinWright    By ColinWright 2021/03/22 @ 12:25:55g -------------------------------- Suppose we use 3^2=9. Find where that's the step we take.        (select only this node)     N_20210322122555f_ColinWright->N_20210322122555g_ColinWright N_20210322122555h_ColinWright    By ColinWright 2021/03/22 @ 12:25:55h -------------------------------- 16 + 9 = 25        (select only this node)     N_20210322122555g_ColinWright->N_20210322122555h_ColinWright N_20210322122555i_ColinWright    By ColinWright 2021/03/22 @ 12:25:55i -------------------------------- 4^2 + 3^2 = 5^2        (select only this node)     N_20210322122555j_ColinWright    By ColinWright 2021/03/22 @ 12:25:55j -------------------------------- So we have generated a Pythagorean triple.        (select only this node)     N_20210322122555i_ColinWright->N_20210322122555j_ColinWright N_20210322122555j_ColinWright->N_20210322120320a_ColinWright N_20210324091207a_ColinWright    By ColinWright 2021/03/24 @ 09:12:07a -------------------------------- Can we generate all Pythagorean Triples using this method?        (select only this node)     N_20210322122555j_ColinWright->N_20210324091207a_ColinWright N_20210324090252a_ColinWright    By ColinWright 2021/03/24 @ 09:02:52a -------------------------------- Find a prime that's one more than a square        (select only this node)     N_20210324090252b_ColinWright    By ColinWright 2021/03/24 @ 09:02:52b -------------------------------- Find another        (select only this node)     N_20210324090252a_ColinWright->N_20210324090252b_ColinWright N_20210324090434a_ColinWright    By ColinWright 2021/03/24 @ 09:04:34a -------------------------------- Find a prime that's the sum of two squares        (select only this node)     N_20210324090252a_ColinWright->N_20210324090434a_ColinWright N_20210324090434d_ColinWright    By ColinWright 2021/03/24 @ 09:04:34d -------------------------------- List in four columns        (select only this node)     N_20210324090434c_ColinWright->N_20210324090434d_ColinWright N_20210322122555b_ColinWright->N_20210322122555d_ColinWright N_20210322122555c_ColinWright    By ColinWright 2021/03/22 @ 12:25:55c -------------------------------- 1, 4, 9, 16, 25, 36, ...        (select only this node)     N_20210322122555b_ColinWright->N_20210322122555c_ColinWright N_20210324090434e_ColinWright    By ColinWright 2021/03/24 @ 09:04:34e -------------------------------- Notice anything?        (select only this node)     N_20210324090434d_ColinWright->N_20210324090434e_ColinWright N_20210322124629c_ColinWright    By ColinWright 2021/03/22 @ 12:46:29c -------------------------------- If you can't then your number is prime.        (select only this node)     N_20210322124629a_ColinWright->N_20210322124629c_ColinWright N_20210322124629b_ColinWright    By ColinWright 2021/03/22 @ 12:46:29b -------------------------------- If you can, and it's AxB with A>1 and B>1, then your number is composite.        (select only this node)     N_20210322124629a_ColinWright->N_20210322124629b_ColinWright N_20210322124629d_ColinWright    By ColinWright 2021/03/22 @ 12:46:29d -------------------------------- This is one way of defining a prime number.        (select only this node)     N_20210322124629c_ColinWright->N_20210322124629d_ColinWright N_20210322124904a_ColinWright    By ColinWright 2021/03/22 @ 12:49:04a -------------------------------- Let's write the numbers in six columns, circle the primes, and see what we get:        (select only this node)     N_20210322124629c_ColinWright->N_20210322124904a_ColinWright N_20210322124629c_ColinWright->N_20210324085639a_ColinWright N_20210322124629e_ColinWright    By ColinWright 2021/03/22 @ 12:46:29e -------------------------------- It's not the only way!        (select only this node)     N_20210322124629d_ColinWright->N_20210322124629e_ColinWright N_20210322124904b_ColinWright    By ColinWright 2021/03/22 @ 12:49:04b -------------------------------- 0 1 [2] [3] 4 [5] 6 [7] 8 9 10 [11] 12 [13] 14 15 16 [17] 18 [19] 20 21 22 [23] 24 25 26 27 28 [29] 30 [31] 32 33 34 35 36 [37] 38 39 40 [41] 42 [43] 44 45 46 [47] ...        (select only this node)     N_20210322124904a_ColinWright->N_20210322124904b_ColinWright N_20210322124904e_ColinWright    By ColinWright 2021/03/22 @ 12:49:04e -------------------------------- How can you explain it?        (select only this node)     N_20210324090515a_ColinWright    By ColinWright 2021/03/24 @ 09:05:15a -------------------------------- This is sort-of related to Pythagoras ... sum of two squares.        (select only this node)     N_20210322124904c_ColinWright    By ColinWright 2021/03/22 @ 12:49:04c -------------------------------- What do you notice?        (select only this node)     N_20210322124904b_ColinWright->N_20210322124904c_ColinWright N_20210322125819a_ColinWright    By ColinWright 2021/03/22 @ 12:58:19a -------------------------------- - - [2] [3] - [5] - [7] - - - [11] - [13] - - - [17] - [19] - - - [23] - ---- - - - [29] - [31] - - - ---- - [37] - - - [41] - [43] - - - [47] ...        (select only this node)     N_20210322124904b_ColinWright->N_20210322125819a_ColinWright N_20210324090708a_ColinWright->N_20210324090434a_ColinWright N_20210324085355a_ColinWright    By ColinWright 2021/03/24 @ 08:53:55a -------------------------------- Find a prime that's one less than a square        (select only this node)     N_20210324085355b_ColinWright    By ColinWright 2021/03/24 @ 08:53:55b -------------------------------- Now find another        (select only this node)     N_20210324085355a_ColinWright->N_20210324085355b_ColinWright N_20210324085541a_ColinWright    By ColinWright 2021/03/24 @ 08:55:41a -------------------------------- When p>3, look at p^2-1        (select only this node)     N_20210324085541b_ColinWright    By ColinWright 2021/03/24 @ 08:55:41b -------------------------------- Do you see anything interesting?        (select only this node)     N_20210324085541a_ColinWright->N_20210324085541b_ColinWright N_20210324085541c_ColinWright    By ColinWright 2021/03/24 @ 08:55:41c -------------------------------- Prove that for p>3, p^2-1 is a multiple of 24.        (select only this node)     N_20210324085541c_ColinWright->N_20210322121359c_ColinWright N_20210324090252c_ColinWright    By ColinWright 2021/03/24 @ 09:02:52c -------------------------------- How many are there?        (select only this node)     N_20210324090252b_ColinWright->N_20210324090252c_ColinWright N_20210324091207b_ColinWright    By ColinWright 2021/03/24 @ 09:12:07b -------------------------------- If so, can you prove it?        (select only this node)     N_20210324091207a_ColinWright->N_20210324091207b_ColinWright N_20210324091207c_ColinWright    By ColinWright 2021/03/24 @ 09:12:07c -------------------------------- If not, can we adapt it?        (select only this node)     N_20210324091207a_ColinWright->N_20210324091207c_ColinWright N_20210322124904c_ColinWright->N_20210322124904e_ColinWright N_20210322124904d_ColinWright    By ColinWright 2021/03/22 @ 12:49:04d -------------------------------- Does this surprise you?        (select only this node)     N_20210322124904c_ColinWright->N_20210322124904d_ColinWright N_20210324085639a_ColinWright->N_20210324090252a_ColinWright N_20210324085639a_ColinWright->N_20210324085541a_ColinWright N_20210324085639a_ColinWright->N_20210324085355a_ColinWright N_20210322122555h_ColinWright->N_20210322122555i_ColinWright N_20210324085541b_ColinWright->N_20210324085541c_ColinWright N_20210324090434a_ColinWright->N_20210324090515a_ColinWright N_20210324090434a_ColinWright->N_20210324090434b_ColinWright