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Catalan’s Conjecture

Although popularly known as Catalan’s conjecture, this is in fact a theorem in number theory, proven by Preda Mihăilescu in 2002, 158 years after it was conjecture in 1844 by French and Belgian mathematician Eugène Charles Catalan.

The theorem states that the only solution in the natural numbers of

{\displaystyle x^{a}-y^{b}=1}

for a,b>1; x,y>0 is x = 3, a=2, y=2, b=3.

Catalan ‘Near Misses’ (i.e. xa – yb <= 10; 2 <= x,y,a,b <= 100; a,b prime)

  • 33 – 52 = 2
  • 27 – 53 = 3
  • 23 – 22 = 62 – 25 = 53 – 112 = 4
  • 25 – 33 = 5
  • 25 – 52 = 42 – 32 = 27 – 112 = 7
  • 42 – 23 = 8
  • 52 – 42 = 62 – 33 = 152 – 63 = 9
  • 133 – 37 = 10

Proving the Conjecture

Similarly to Fermat’s last theorem, the solution to this conjecture was assembled over a long period of time. In 1850, Victor Lebesgue established that b cannot equal 2. But then it was only after around 100 years, in 1960, that Chao Ko constructed a proof for the other quadratic case: a cannot equal 2 unless x = 3.

Hence, this left a,b odd primes. Expressing the equation as (x-1)(xa-1)/(x-1) = yb one can show that the greatest common divisor of the two left hand factors is either 1 or a. The case where the gcd = 1 was eliminated by J.W.S Cassels in 1960. Hence, only the case where gcd = a remained.

It was this “last, formidable, hurdle” that Mihailescu surmounted. In 2000, he showed that a and b would have to be a ‘Wieferich pair’, i.e. they would have to satisfy ab-1 ≡ 1 (mod b2) and ba-1 ≡ 1 (mod a2). In 2002,  he showed that such solutions were impossible!

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Surreal Numbers

Surreal numbers were first invented by John Horton Conway in 1969, but was introduced to the public in 1974 by Donald Knuth through his book ‘Surreal Numners: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness‘.

What are Surreal Numbers?

Surreal numbers are the ‘most natural’ collection of numbers that include both real numbers and the infinite ordinal numbers of Georg Cantor. The surreals have many of the same properties as the reals, including the usual arithmetic operations. Hence, they form an ordered field.

For a surreal number x we write x = {XL|XR} and call XL and XR the left and right set of x,respectively. These will be explained below.

Conway Construction

“Surreal numbers are constructed inductively as equivalence classes of pairs of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set.”

– Wikipedia

Using the Conway construction, we construct the surreal numbers in stages along with an ordering ≤ such that for any two surreal numbers a and b either a ≤ b or b ≤ a.

Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. Therefore, if x = {XL|XR} then for each xL ∈ XL and xR ∈ XR, xL is not greater than xR.

In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: { | } = 0.

Subsequent stages yield the following:

  • {0|} = 1, {1|} = 2, {2|} = 3, etc;
  • {|0} = -1, {|1} = -2, {|2} = -3, etc.

For more, click here.

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VIDEO: Where Algorithms Fail

My sister showed me this video a few days ago and I’d thought I’d share it with you. In it Cathy O’Neil, author of the new book ‘Weapons of Math Destruction‘, discusses the danger of algorithms, giving a few examples to illustrate her eye-opening points. O’Neil then goes on to highlight a few steps we can take, as a society, in order to overcome these pitfalls. I found this short video thoroughly interesting and hope you enjoy it as well!

“Algorithms decide who gets a loan, who gets a job interview, who gets insurance and much more — but they don’t automatically make things fair. Mathematician and data scientist Cathy O’Neil coined a term for algorithms that are secret, important and harmful: “weapons of math destruction.” Learn more about the hidden agendas behind the formulas.”

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VIDEO: Napkin Ring Problem

If you were to core a sphere (remove a cylinder from it), you are left with a shape that looks like a napkin ring. This is a “bizarre” shape, as if you have two napkin rings with the same height, they will have the same volume regardless of the size of the initial spheres that they came from. How do you prove this?

Here’s a few hints to try and solve it yourself before watching the Vsauce video below which reveals the answer:

  • There are a few variables that need to be found: the height of the napkin ring, the radius of the starting sphere and the radius of the cylinder. Using these variables you can find a volume equation.
  • You don’t need to find the volume of the whole napkin ring in one go. This is because, as the two napkin rings have to be the same height, it’s enough to show that any slice of the napkin rings has to have the same area. If every pair of slices has the same area, then the napkin rings have the same volume.

Solution:

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MATHS BITE: The Kolakoski Sequence

The Kolakoski sequence is an infinite sequence of symbols {1,2} that is its own “run-length encoding“. It is named after mathematician Willian Kolakoski who described it in 1965, but further research shows that it was first discussed by Rufus Oldenburger in 1939.

This self-describing sequence consists of blocks of single and double 1s and 2s. Each block contains digits that are different from the digit in the preceding block.

To construct the sequence, start with 1. This means that the next block is of length 1. So we require that the next block is 2, giving the sequence 1, 2. Continuing this infinitely gives us the Kolakoski sequence: 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, etc.

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VIDEO: Spiral Sculptures

John Edmark is an artist and professor at Stanford University who has used the Golden Angle to sculpt spirals. The Golden Angle is derived from the Golden Ratio: it is the smaller of the two angles created by dividing the circumference of a circle according to the golden ratio and comes out to be around 137.5°.

For more click here.

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VIDEO: Connecting the Drops

Today I thought I’d share a video that I came across the other day. Hope you enjoy!

“Bacteria and viruses hitch a ride inside droplets of all kinds—sneezes, raindrops, toilet splatter. By reviewing footage of different types of drops, applied mathematician Lydia Bourouiba records and measures where they disperse in order to better understand how diseases spread. Watch how Bourouiba designs tests—some inescapably humorous and awkward—to study infectious disease transmission.”

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