Der erste Shalan-Satz stellt eine ungeheure Vereinfachung dar, aber kaum wen interessiert's, obwohl man ALLE Pythagoräischen Tripel findet (siehe QP): Aargghh!
#SharingIsCaring #sharingisthenewlearning #education #mathematics #maths #geometry #Mathe #Pythagoras #Pythagorastheorem #SatzdesPythagoras
My screenshot reads: "Pythagoras theorem: a²+b²=c² First Shalan theorem 0.5d(m²-1) as "a", dm as "b", 0.5d(m²+1) as "c" = a+d (0.5d(m²-1))²+d²m²=(0.5d(m²+1))² =(a+d)² Start with any natural number greater than two. Find its factors, fractions if necessary (24 is 18 times 4/3; the first Shalan theorem yields 7²+24²=(7+18)²). Define one of the factors as "d," the difference between a and c, the other as "m". If only one of the factors is odd, it must serve as m, because d must then be even. If the number is even, two is the smallest possible difference; if the number is odd, 1 is the smallest possible difference. An odd number could be the hypotenuse of a Pythagorean triple too: (17=2(4²+1)/2)=>the corresponding primitive triple is 8,15,17. The effectiveness of the formula can be demonstrated using the example of the number 8432. Its divisors are 1 and 8432 (not relevant here), 2, 4, 8, 16, 17, 31, 34, 62, 68, 124, 136, 248, 272, 496, 527, 1054, 2108, 4216."
My screenshot reads: "d=2, m=4216: 17774655²+8432²=17774657² d=4,m=2108: 8887326²+8432²=8887330² d=8,m=1054: 4443660²+8432²=4443668² d=16, m=527: 2221824²+8432²=2221840² d=34,m=248: 1045551²+8432²=1045585² d=62, m=136: 573345²+8432²=573407² d=68,m=124: 522750²+8432²=522818² d=124,m=68: 286626²+8432²=286750² d=136,m=62: 2613242+8432²=261460² d=248,m=34: 143220²+8432²=143468² d=272,m=31: 130560²+8432²=130832² d=496,m=17: 71424²+8432²=71920² d=1054,m=8: 33201²+8432²=34255² d=2108,m=4: 15810²+8432²=15810² d=4232,m=2: 6324²+8432²=10540²"
!!!SENSATION!!!
BREAKING: How To #Pythagoras 2.0
#Pythagorastheorem vs 1st #Shalantheorem, which is lazybones's darling because it requires a single number and squaring of one number that is at most as large as the starting number!
#sharingisthenewlearning #education #mathematics #maths #geometry
A formula that describes the famous Yang Hui or Pascal's Triangle If you enlarge a square, you need two new rectangles of the same size and a square. The area of these rectangles and the square must be a square number, if you want to find the Pythgorean triples ab,c with a²+b²=c²
Visualization of the Yang Hui or Pascal's Triangle, dimensions 0, point, 1, line, 2, square, 3, cube, with LEGOs. Blue: the starting bodies white, gray, red, black: the components that are added, for example two rectangles and a square or three wall elements, three edge elements and a cube To find any Pythagorean triple 2025 belongs to, find its divisors: 2025=45²=3⁴*5² =1*2025 =3*675 =5*405 =9*225 =15*1350 =25*81 =27*75 =45*45 =75*27 =81*25 =135*15 =225*9 =405*5 =675*3 There are 14 combinations, therefore 2025 belongs to 14 Pythagorean triples. If d=1, b=2ad+d², then 2a+1= 2025 2025-1=2a, a=1012, b²=2025, c=1013: => 1012²+45²=1013² works without multiplikations! This is a primitive Pythagorean triple.
Hi fam,
have a fulfilling year 45² that equals 2025!
2025 is a square number and belongs to 14 Pythagorean triples.
(d(m²-1)/2)²+d²m²=(d(m²+1)/2)²
d=1, m=2025
2050312²+2025²=2050313²
CHEAP TRICK: ALT: 1012²+2025=1013²
#Pythagorastheorem #1stShalantheorem #mathematics
“How The Square Root Of 2 Became A Number”, Quanta (www.quantamagazine.org/how-the-squa...).
On HN: news.ycombinator.com/item?id=4075...
#Numbers #IrrationalNumbers #Mathematics #Maths #PythagorasTheorem #Dedekind #Cantor #History #Reals #Irrationals
