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Paul Arnheim-Projekt

@paula-projekt.bsky.social

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

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.

January 2, 2025 at 2:40 AM

Ranjit Mathew

@rmathew4bs.bsky.social

“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

How the Square Root of 2 Became a Number | Quanta Magazine

Useful mathematical concepts, like the number line, can linger for millennia before they are rigorously defined.

www.quantamagazine.org

June 22, 2024 at 10:55 AM

Paul Arnheim-Projekt

@paula-projekt.bsky.social

!!!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

$My screenshot reads: "Pythagoras theorem: a²+b²=c² First Shalan theorem 0.5d(m²-1) as "a", dm as "b", 0.5d(m²+1) as "c" = a+d (0.5d(m²-1))²+d²m²=(0.5d(m²+1))² =(a+d)² Start with any natural number greater than two. Find its factors, fractions if necessary (24 is 18 times 4/3; the first Shalan theorem yields 7²+24²=(7+18)²). Define one of the factors as "d," the difference between a and c, the other as "m". If only one of the factors is odd, it must serve as m, because d must then be even. If the number is even, two is the smallest possible difference; if the number is odd, 1 is the smallest possible difference. An odd number could be the hypotenuse of a Pythagorean triple too: (17=2(4²+1)/2)=>the corresponding primitive triple is 8,15,17. The effectiveness of the formula can be demonstrated using the example of the number 8432. Its divisors are 1 and 8432 (not relevant here), 2, 4, 8, 16, 17, 31, 34, 62, 68, 124, 136, 248, 272, 496, 527, 1054, 2108, 4216."$

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"

June 24, 2025 at 12:46 PM

Jon @ The Maths Vault

@mathsvault.bsky.social

Plenary — Pythagoras theorem

#maths #mathsteacher #UKMathsChat #GCSEMaths #Plenary #MathsDiscussion #TalkTasks #PythagorasTheorem #Pythagoras #Geometry

May 20, 2026 at 7:30 AM

Jon @ The Maths Vault

@mathsvault.bsky.social

Discuss — Pythagoras Theorem

#maths #mathsteacher #UKMathsChat #GCSEMaths #MathsChat #Discussion #RealLifeMaths #Applications #PythagorasTheorem #Pythagoras #Geometry #Ratio #Proportion