# Triangles and Rectangles

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Consider the following question by Bill Sands (asked in 1995):

Are there right triangles with integer sides and area, associated with rectangles having the same perimeter and area?

Try to test your intuition. The solution to this problem is NOT so simple. The solution was published by Richard K. Guy in 1995: http://www.jstor.org/stable/2974502

If you have a simpler solution, please write it in a comment below. Even I don’t understand much of the solution.

# Under 40

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The age of 40 is considered special in mathematics because it’s an ad-hoc criterion for deciding whether a mathematician is young or old. This idea has been well established by the under-40 rule for Fields Medal, based on Fields‘ desire that:

while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others

Though it must be noted that this criterion doesn’t claim that after 40 mathematicians are not productive (example: Yitang Zhang).  So I wanted to write a bit about the under 40 leading number theorists which I am aware of (in order of decreasing age):

• Sophie Morel: The area of mathematics in which Morel has already made contributions is called the Langlands program, initiated by Robert Langlands. Langlands brought together two fields, number theory and representation theory. Morel has made significant advances in building that bridge. “It’s an extremely exciting area of mathematics,” Gross says, “and it requires a vast background of knowledge because you have to know both subjects plus algebraic geometry.” [source]
• Melanie Wood: Profiled at age 17 as “The Girl Who Loved Math” by Discover magazine, Wood has a prodigious list of successes. The main focus of her research is in number theory and algebraic geometry, but it also involves work in probability, additive combinatorics, random groups, and algebraic topology.  [source1, source2]
• James Maynard:  James is primarily interested in classical number theory, in particular, the distribution of prime numbers. His research focuses on using tools from analytic number theory, particularly sieve methods, to study primes.  He has established the sensational result that the gap between two consecutive primes is no more than 600 infinitely often. [source1, source2]
• Peter Scholze: Scholze began doing research in the field of arithmetic geometry, which uses geometric tools to understand whole-number solutions to polynomial equations that involve only numbers, variables and exponents. Scholze’s key innovation — a class of fractal structures he calls perfectoid spaces — is only a few years old, but it already has far-reaching ramifications in the field of arithmetic geometry, where number theory and geometry come together. Scholze’s work has a prescient quality, Weinstein said. “He can see the developments before they even begin.” [source]

# Prime Consequences

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Most of us are aware of the following consequence of Fundamental Theorem of Arithmetic:

There are infinitely many prime numbers.

The classic proof by Euclid is easy to follow. But I wanted to share the following two analytic equivalents (infinite series and infinite products) of the above purely arithmetical statement:

• $\displaystyle{\sum_{p}\frac{1}{p}}$   diverges.

For proof, refer to this discussion: https://math.stackexchange.com/q/361308/214604

• $\displaystyle{\sum_{n=1}^\infty \frac{1}{n^{s}} = \prod_p\left(1-\frac{1}{p^s}\right)^{-1}}$, where $s$ is any complex number with $\text{Re}(s)>1$.

The outline of proof,   when $s$ is a real number, has been discussed here: http://mathworld.wolfram.com/EulerProduct.html

# Arithmetic Puzzle

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Following is a very common arithmetic puzzle that you may have encountered as a child:

Express any whole number $n$ using the number 2 precisely four times and using only well-known mathematical symbols.

This puzzle has been discussed on pp. 172 of Graham Farmelo’s “The Strangest Man“, and how Paul Dirac solved it by using his knowledge of “well-known mathematical symbols”:

$\displaystyle{n = -\log_{2}\left(\log_{2}\left(2^{2^{-n}}\right)\right) = -\log_{2}\left(\log_{2}\left(\underbrace{\sqrt{\sqrt{\ldots\sqrt{2}}}}_\text{n times}\right)\right)}$

This is an example of thinking out of the box, enabling you to write any number using only three/four 2s. Though, using a transcendental function to solve an elementary problem may appear like an overkill.  But, building upon such ideas we can try to tackle the general problem, like the “four fours puzzle“.

This post on Puzzling.SE describes usage of following formula consisting of  trigonometric operation $\cos(\arctan(x)) = \frac{1}{\sqrt{1+x^2}}$ and $\tan(\arcsin(x))=\frac{x}{\sqrt{1-x^2}}$ to obtain the square root of any rational number from 0:

$\displaystyle{\tan\left(\arcsin\left(\cos\left(\arctan\left(\cos\left(\arctan\left(\sqrt{n}\right)\right)\right)\right)\right)\right)=\sqrt{n+1}}$.

Using this we can write $n$ using two 2s:

$\displaystyle{n = (\underbrace{\tan\arcsin\cos\arctan\cos\arctan}_{n-4\text{ times}}\,2)^2}$

or even with only one 2:

$\displaystyle{n = \underbrace{\tan\arcsin\cos\arctan\cos\arctan}_{n^2-4\text{ times}}\,2}$

# Intra-mathematical Dependencies

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Recently I completed all of my undergraduate level maths courses, so wanted to sum up my understanding of mathematics in the following dependency diagram:

I imagine this like a wall, where each topic is a brick. You can bake different bricks at different times (i.e. follow your curriculum to learn these topics), but finally, this is how they should be arranged (in my opinion) to get the best possible understanding of mathematics.

As of now, I have an “elementary” knowledge of Set Theory, Algebra, Analysis, Topology, Geometry, Probability Theory, Combinatorics and Arithmetic. Unfortunately, in India, there are no undergraduate level courses in Mathematical Logic and Category Theory.

This post can be seen as a sequel of my “Mathematical Relations” post.