For years, mathematicians have worked to demonstrate that x3+y3+z3 = k, where k is defined as the numbers from 1 to 100. This theory is true in all cases except for an unproven exception: 42.
By 2016 and over a million hours of computation later, researchers of the UK’s Advanced Computing Research Center had its solution for 42.
More intriguing number facts here.
The three blue points always lie on a straight line. The blue points are the closest points to the moving red point on the lines. In other words the blue points are the projections of the moving red point to the lines.
Summation of Alternating Inverse Powers of Phi…
List of sums of reciprocals.
The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.
Clever visual proof by Mike Hirschhorn.
“Euler’s line” (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).
Other notable points that lie on the Euler line include the de Longchamps point, the Schiffler point, the Exeter point, and the Gossard perspector.
If you place squares on the sides of any parallelogram, their centers will always form a square.
A visual intuitive proof that √ab cannot be larger than (a+b)/2, where a, b ∈ R*+
Continue reading “The Arithmetic-Geometric Mean Inequality”
No matter how you choose to place the red points on the circumference, the three blue points will lie on a straight line (shown in red). This works for any conic (which may be a circle, ellipse, parabola or hyperbola).
It is conjectured that n is a sum of 3 cubes if n is a number that is not congruent to 4 or 5 mod 9. The number 33 enters this category, but for 64 years no solutions emerged — that is, whether the equation 33 = x³ + y³ + z³ has an integer solution. Continue reading ““Stubborn” Number 33″